The inversion of $n imes n$ matrices which are a sum of a Toeplitz and a Hankel matrix (briefly, T+H matrices) has been studied for a long time. It is known that the inverses of (nonsingular) T+H matrices are also matrices with a special structure, so-called T+H Bezoutians. The converse problem, namely the inversion of T+H Bezoutians has been given little attention. In this talk I will present some of the recent results concerning the inversion of centrosymmetric or centroskewsymmetric T+H Bezoutians. Open problems will also be mentioned.