I will start with a brief overview of massive, bi- and multi-metric theories of gravity and their cosmological implications. Bimetric gravity, despite its theoretically strong motivations and cosmologically fascinating implications, in particular as an alternative explanation of the late-time acceleration of the Universe, has been widely argued to be plagued by severe (ghost and/or gradient instability) problems. I will argue in this talk that there are several possibilities to save the theory and obtain healthy cosmological solutions with implications which may or may not differ significantly from the standard cosmological picture when the predictions of the theory are compared to observations. I will illustrate that in either case the underlying physical mechanism behind the cosmic acceleration is theoretically of a completely different nature which provides a neat solution to the new cosmological constant (dark energy) problem in a technically natural way.