A basic question in the theory of quasiregular maps has been the following existence problem: Given two oriented and connected Riemannian n-manifolds M and N, does there exist a nonconstant quasiregular map of M into N? The interesting case is when M is noncompact. In particular, if M=R^n and such a map exists, Gromov calls N quasiregularly elliptic. We will consider the quasiregular ellipticity problem for N closed. The case n=2 is classical and the case n=3 is pretty well understood in terms of the fundamental group. Apart from some special examples not much was known until recently for n>3. A break-through result was obtained by M. Bonk and J. Heinonen in 2001 when they proved that the dimension of the de Rham cohomology ring of N has an upper bound depending only on n and K for a K-quasiregular map of R^n into N. In the talk various questions related to the quasiregular ellipticity problem are discussed including the ideas in the proof of the theorem of Bonk and Heinonen.