Making the transition from Classic to Impossible can be a steep climb. Here is a comparision of the main differences between both difficulty modes.

Tactical

XCOM soldiers start with base 3 HPs (and +1 from Body Armor), a reduction that will have effects during all missions but making them very susceptible to KIAs from Sectoid Plasma Pistols at game start.
The aliens will deploy on bigger numbers (usually 1 additional pod) during the 1st mission and specially on Terror Sites/UFOs. It can be a challenge to know how many pods are still inactivated.
The AI will use grenades more, specially when facing Muton Elites or Heavy Floaters. After the aliens start fielding grenades the full cover positions in most maps will be very risky to use, since it will be blown away.

General Warnings

Missions at the beginning of each month are the most dangerous since usually new aliens will be introduced. April will bring Floaters, Mutons (initially equipping with LPR) in May, Cyberdiscs come in June and Beserkers in July.
Expect several of the alien heavy units on the same mission. Terror Sites with 5 Sectopods can be pretty terrifying.
Staying out of enemy Line of Sight (LOS) is crucial for the survival of XCOM soldiers. If they are seen they will be fired upon, grenaded and poisoned, and usually killed by a single plasma shot, specially in the first months.
During Abductions/Terror Sites/UFO missions the aliens will have a number of packs tied to certain locations while others will be patrolling across the entire map. During the beginning of the mission and depending on the map you may expect to have those packs activated right in front of you.
When there are no active aliens, good map knowledge is crucial to know the possible locations of inactive alien packs.

Panic Management

It is possible, if exeedingly difficult, to last out the first month (and hopefuly, a campaign) without losing any countries in Impossible. Required is luck, extreme skill and knowledge of the game, and probably a lot of save-state abuse.

You starting country; either Africa for the starting money, or Europe for cheaper Workshops.
- Alternatively, if you're just out for not losing the Contienent Bonuses and don't mind one or two losses; start in Asia or Europe, and let them take a hit.
The biggest threat is, naturally, the two Abduction missions: Since Asia and Europe have 4 countries each, any passed Abduction mission each means 6 panic total, while South America with 2 countries can suck it up untill you get your sattelites in.
Normally, with those two Missions, and helping one country each, that means (with luck) 4 countries exactly that'll be at full panic by the end of the month

The key is the Council Mission/Request, that needs to happen before the 16th

Option A: A mission that includes Panic Reduction (and a significant amount of cash) in an at-risk nation
- panic reduction in withdrawing country, 3 left, one Uplink, get requisite 10 engineers before 16th, build 2nd Uplink, 3 satts for the 3 remaining countries
flexibility to let second Abduction Mission be for cash or scientists
Option B: A mission or a request that gives a strong number of engineers
- both abductions with max engies, need 20, either build 1 generator (or cheat with Difficulty Change, I guess- doing this on EW to unlock last Second Wave options--- woulda been nice if they carried over), so you can have two Uplinks, thus up to 6 Satts, and start April well-off.

Either way, you'll be selling most corpses, and should dump elerium into grey market
danger; your research will be slow as consequence (and thus, your armor/weapons R&D), particularly in option B

(In Progress, just saving my spot while I gather my thoughts and continue, format a little better)--Xuncu 03:24, 3 December 2013 (EST)

Know The Map

Some maps are specially dangerous on Abduction missions, due to their openness, small size and/or general lack of full cover:

Other Abduction maps are more favorable, with much better cover, more movement options and/or easy roof access.

Take Cover

Hunkering behind full cover is the key to survive, specially on the first missions.
If your soldiers in full cover can be seen by the aliens, hunker down. It will give them a total of +80 Defense and reduce a Sectoid's chance of hitting to
Only go on Move, overwatch or Fire if that soldier can survive the return fire from any active aliens with LOS on the soldier.
Learn which cover cannot be removed by alien gunfire or explosives. Destructible cover can give a very wrong sense of protection and get soldiers killed.
Pull back whenever outnumbered, specially right after two or three alien packs activate all of a sudden.
If your rookie is feeling brave and sure of hitting with 65% odds, consider what may happen when that alien(s) is still alive after he/she misses:
1. A single Alien that needs 2 shots to kill one of your soldiers is a nuisance.
2. 2 or more Aliens that need 2 shots to kill one of your soldiers are a threat. The same for 1 Alien that only needs 1 shot and has your soldier on its sights.
3. 2 or more Aliens that need only 1 shot to kill a soldier is risking disaster for the entire squad.

Move Softly

When scouting with a unit starting on a location in full cover, don't advance to the maximum of moving range during the 1st Action. That way you'll be able to go back, not to your starting location in view of any aliens, but to a square next to it, that prevents the aliens to have Line of Sight in your unit.
If choosing between spending the 2nd action to move to full cover or to hunker down behind half cover, go for the latter. Hunkering down gives you +40 Defense (like full cover) and prevents critical hits (unlike full cover).
Move very, very slowly when you're sure to approach the location of aliens, either active or inactive. In many occasions it is better to only have 1 unit moving on each turn. This allows to have the entire squad prepared to deal with any hostile encounters.
Roofs are essential to move without alien interference. Be aware that some alien packs may be waiting for activation on roofs, though.
Aliens overwatching are more deadly: a Sectoid can kill a Rookie when firing on Overwatch, and LPRs can kill Sergeants, if they are not carrying vests.
Advancing through dashing will usually result in:
1. The activation of further alien packs.
2. Your soldier missing the shot and/or not killing the alien.
3. Your soldier being killed by the reaction of any active aliens.
4. All of the above.

Strike Efficiently

When a soldier reaches a full cover position where it is overflanking an alien but the odds don't ensure a kill, Hunker Down or go on Overwatch. With the last option, the alien will usually 'freeze' (technically this is a bug/exploit and doesn't work 100%) and will not take any action during its turn, as long as all XCOM units on its LOS are in full cover.
Cars will blow up when hit by a Rocket or two Frag Grenades and add an additional 6 Damage points to any aliens huddled against them.
Besides the Damage they inflict, Frag Grenades and Rockets can be used to remove any cover and leave the aliens exposed. Firing at an unit that just got exposed by an explosion gives the attacker a +20/40 Aim boost (depending whether the alien was behind half or full cover), plus a +50% bigger chance to inflict a Critical Hit.
One key ability is to be able to kill all or most of a alien pack before or when it's activated, through Rockets, Grenades or Overwatch shots.
The longer you take to kill the last active alien, the likelier the odds of an passive pack being activated and forcing you to pull back.
Keep a close eye on the squad's ammo levels. If a soldier or the squad is not engaged in combat have all of them reload their weapons. Reload to 100% ammo before activating an alien pack.

Squad Equiping

When equipping during the initial months, the choice is between grenades or Nano-Fiber Vests. The first means the ability to remove cover, and weaken/kill single or clustered aliens. The second the ability to survive a single Plasma Pistol or LPR shot.
Completing the Slingshot Missions not only gives you early access to the OTS but also to a Lieutenant Heavy and the ability to capture the Plasma Rifle from the Mutons that appear during the missions. You can also capture a Thin Man you get his LPR which allow you to skip the Phoenix Cannon and Laser Cannon completely by going straight to the Plasma Cannon – Just remember that none of the missions give Panic Reduction as a reward, and the rewards are static and not affected by the Second Wave option High Stakes.
1. Zhang is considerably better than any other heavy of his rank with his health surpassing that of a Colonel Heavy.

Classes

The Heavy is the best starting class because of the Rocket Launcher and the Bullet Swarm ability.

The Rocket Launcher is a life saver during the first months due to its ability to kill clustered aliens with 1 shot. By adding Shredder Rocket each Heavy can carry the equivalent of two grenades, with even better range and damage abilities.
Bullet Swarm is arguably the best second best perk at the beginning. It allows the Heavy to Fire and Hunker Down in the same turn, protecting it from alien counterfire.

Snipers are also very useful if they have Squad Sight, allowing them to fire without being exposed to the aliens.
Assaults need to be carefully used. Run & Gun allows to outflank and overwatch aliens but long dashes can provoke the activation of more aliens.
Supports are underpowered until Laser Rifles or LPRs are available.

The Assault Rifle has a limited damage and crit ability compared to the other classes weapons.
The Smoke Grenade can only be used once or twice (if they have Smoke & Mirrors]] and doesn't give the same protection (+20 Defense) as hunkering down (double, or +40 behind full cover]].
Sprinter is useful for spotting aliens and running away from trouble.

Aliens

Game Start

Sectoid HPs are increased to 4 HPs, preventing them from being killed by single Frag Grenades, although a well placed frag can tale out the cover of three Sectoids, injure them and then leave them open to the rest of your squad
Floaters and Thin Men HPs are increased from 4 to 6, making Assault Rifles even more unreliable to kill them, however, their Aim remains the same as in Classic. Rocket Launchers still retain their 'kill with 1 shot' ability against both aliens.
Outsiders have 90 Aim, 10 Defense and 17 Mobility (the same as Bezerkers). Be very careful with activating Outsiders on Impossible by mistake - a single one on the loose will wreck havoc amongst your troops with their high Aim and Movement.
Chryssalids remain the same as Classic but Zombies are the bad news. They have 14 HPs (10 in Classic) and their Damage bonus is increased by 4 (+2 on Classic).

Mid Game

Muton and Cyberdisk stats remain the same as in Classic.
Drones HP is increased to 7 HPs on Impossible, 2 more than in Classic, turning them into a bigger nuisance. It also prevents Drones from being killed in 1 shot by suppressing them with a Heavy Plasma with HEAT Ammo and Mayhem.
Sectoid Commanders HP is increased from 10 to 14 HPs on Impossible. They also have a 35 Will bonus (25 on Classic), allowing them even better odds on Mind Control attempts.
Bezerkers health is upgraded 25% from 20 HPs to 25 and they deal 2 extra points of Damage on Impossible (for a total of 11 base damage).

Late Game

Muton Elites have +10 Aim (for a total of 100 Aim on Impossible), giving the best accuracy on the game. In addition, their health is increased on Impossible from 14 to 18 HPs, while their Defense remains at +30. This combination of high stats, their Heavy Plasmas and their numbers (just facing 3 can be a nightmare) makes them the most dangerous aliens on all situations.
Heavy Floaters health is added 2 HPs to a total of 16. Their mobility and bigger usage of Alien Grenades (3 tossing grenades at one soldier with deal 15 points of damage) makes them also very dangerous.
Both Elite Mutons and Heavy Floaters now can't be killed with a single plasma shot that isn't ability enhanced (i.e. Headshot, Shredder, Bring Them On) since it will deal a maximum of 15 damage with a critical hit.
Ethereals have +5 HPs and a 35 Will bonus (25 on Classic), allowing them better odds on Mind Control attempts.
Sectopods stats remain the same as in Classic. However, you'll face more of them during missions instead of just 1. Sectopods also seem to use more their Cannon to blow terrain.

Enemy Within

Among the known additions/modifcations that the Enemy Within DLC will bring, is that both Classic and Impossible will both receive upgrades in difficulty. This may be simply additive, or will scale (ie; for however much harder Classic becomes, if Impossible become moreso by the same, or a multiplied amount).
Alien squad numbers may also change, either in number of squads, or if aliens will deploy in squads of more than three.
If Impossible Terror Missions will still be mostly Sectopods: with the Second Wave option; "Training Roulette", it may be possible to build a balanced squad of soldiers-- that could all have the HEAT Ammo ability.

Impossible Difficulty (EU2012)

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