We present some new results and question related to the Center-Focus problem for Abel differential equation. There was recently a serious progress in understanding some aspects of this problem. In particular, F. Pakovich and M. Muzychuk completely solved the vanishing problem for polynomial moments: this is an infinitesimal version of the Center-Focus problem. On the other hand, some initial results on the "Poincare inversion problem" have been obtained. This problem asks for a characterization of all possible sequences of Taylor coefficients of the Poincare first return mapping, and for reconstruction of the original Abel equation from the given sequence of its Poincare coefficients. This generalizes the classical C-F problem (where all the coefficients besides the first one are zero). In the Poincare inversion problem, besides the usual "Composition condition" which is a conjectural Center condition, some other types of compositions arise. Closed connections have been found also with the classical Moment inversion problem (which naturally appears as the infinitesimal version of the Poincare inversion). Very encouraging connections have also been found with the "algebraic sampling" problem (i.e. the problem of reconstructing a non-linear model from a set of measurements), as it appears in Signal Processing.