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Generally used only for special situations, heavy weapons can have a variety of effects. They use their own category of ammo called Power Cells, which is harder to come across in the game, so make sure to use your heavy weapons only when you need to.
M-100 Grenade Launcher: Found early in the game as part of the story. This weapon can put a lot of heart on multiple enemies, but it is hard to angle the shots correctly. It is often left behind because there are other heavy weapons which are better suited for any situation you might want the grenade launcher for.
ML-77 Missile Launcher: A great weapon that fires guided missiles. Very effective slow-moving armored bosses like Heavy Mechsand Scions, not as great against weak, fast moving opponents. Can be obtained via a research project.
M-622 Avalanche: This weapon shoots Cryo rounds which freeze targets. It can be very effective, particularly on levels where you face hoards of fast-moving Husks, but those who use Cryo Ammo will find it a little redundant. Obtained via research.
M-920 Cain: A freaking nuke launcher. What do you expect? It does a ton of damage but is hampered by slow firing speed and by ammo usage. Its great for clearing blobs of enemies and pretty good against bosses as well. It is obtained via research.
Collector Particle Beam: Gained in a mid-game story mission, the Particle Beam is extremely accurate and does a lot of damage against shields and biotics but not that great against armor. It is good against almost all Infantry and can vaporize an un-armored opponent within a second or two. This is my favorite heavy weapon, although it is less interesting if your character does not already carry another weapon which is good against armor.