Here is the full list of all 42 Anomaly: Warzone Earth achievements. It takes around 25-30 hours to unlock all of the achievements on Windows.
Complete Tokyo chapter of Story Mode
Destroy every tower type 10 times each
Use 4 different abilities simultaneously
Use every ability 10 times each
Use one Smoke Screen ability to protect 6 units
Complete Baghdad chapter of Story Mode
Protect deployed Decoy with Smoke Screen
Collect 14 Carusaurum caches
Destroy 10 towers
Collect 20 abilities
Earn a gold medal
Repair 5 units
Fully upgrade a unit
Fully upgrade every unit type 2 times each
Earn 50.000 $
Use one Repair ability to fix 400 damage
Use one Airstrike ability to kill 4 towers
Complete Story Mode
Complete Baghdad Mayhem
Complete Tokyo Raid
Earn 10 gold Directness medals
Earn 5 gold Ruthlessness medals
Earn 10 gold Efficiency medals
Earn every gold medal during Story Mode
Earn every Squad Assault Mode gold medal
Destroy 5000 towers
Complete a mission with squad composed of 6 different units
Complete a mission without using Repair ability
Complete mission 1 having 7 Repair ability charges in Advanced difficulty mode
Complete mission 2 with every tower killed in Advanced difficulty mode
Complete mission 3 while moving your squad in front each Scorcher in Advanced difficulty mode
Complete mission 4 without using Decoy ability in Advanced difficulty mode
Complete mission 5 having 20 ability charges in Advanced difficulty mode
Complete mission 6 while beating timed part in less than 190 seconds in Advanced difficulty mode
Complete mission 7 without a single unit upgrade in Advanced difficulty mode
Complete mission 8 with transporter not being hit in Advanced difficulty mode
Complete mission 9 without Hero getting knocked down in Advanced difficulty mode
Complete mission 10 without using Air Strike ability in Advanced difficulty mode
Complete mission 11 without letting Energizer towers respawn other towers in Advanced difficulty mode
Complete mission 12 while killing at most 44 towers in Advanced difficulty mode
Complete mission 13 while letting at least 7 artillery strikes hit your squad in Advanced difficulty mode
Complete the last Boss stage while allowing at most 6 respawn waves in Advanced difficulty mode