An open Riemann surface is called parabolic in case every bounded subharmonic function on it reduces to a constant. Several authors introduced seemingly different analogs of this notion for complex manifolds of arbitrary dimension. In the first part of this expository talk, I will compile these notions of parabolicity, compare them and look at some examples. Then I will relate some of these notions to the linear topological type of the Fréchet space of global analytic functions on the given Stein manifold. This will allow us to look at these notions from functional analysis point of view. Finally, I will consider ''polynomials'' on S-parabolic manifolds and report on some general results about these objects. Most of what I will report in this talk is from joint work with A. Sadullaev.