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A complex manifold $M$ is said to be $S^*$-parabolic if it possesses a continuous plurisubharmonic function $\Phi$ that is maximal outside a compact subset of $M$. In analogy with $(\mathbb{C}^n, log \|z\|)$, one defines ($\Phi$)-polynomials as analytic functions $f$ on $M$ with the property that there exist positive integers $d$ and $c$ such that $|f(z)|\leq d\Phi + c$ for all $z\in M$. In the first part of the talk, we will review different notions of parabolicity for complex manifolds and look at them from a functional analysis point of view. In the second part of the talk, we will discuss polynomials on $S^*$-parabolic manifolds. Most of what I will report in this talk is joint work with A. Sadullaev.