How to Build Your Own Acoustic Panels
Sound Panels can be shockingly expensive to purchase and some “home remedies” like egg crates and moving blankets really don’t do much for cutting down errant sound waves when working in an open studio space or for a mixing room. Fortunately there is a moderate priced DIY method of cutting down room echoes and reverb:
Suitable applications for sound absorption panels include:
Reducing slap-back echo and reverberation in large halls, auditoriums, and contemporary churches
Reducing chatter, din, and noise in crowded gathering spaces, restaurants, or bars
Providing more accurate listening conditions in recording studios and control rooms
AcousticsFreq.com | Read the Full Article

This is a really thorough article and a fairly affordable project to undertake. However, it’s best if you own or have title to the room you’re installing panels. Unless there’s a minimally invasive way to do this for an apartment, some people may be left in the cold.
If you can build this in an apartment, hanging them should be no more destructive to the walls than hanging a large picture.