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I need to create a body, which reacting with another body. I need this body to behave like it is background (another bodies should pass through this body), but I also need body to trigger callback functions. Can I do this?

Yes. Simplest way is using Fixture:setSensor(true). The "background object" won't collide with anything, but it will throw callbacks.

A more complex way to do it would be with Fixture:setFilterData. With that function you can make the background body collide with some objects but not with others, make groups, etc.

Yes, it it exactly what I want! Thanks a lot!