Abstract. We describe all proper holomorphic mappings of the symmetrized polydisc and study its geometric properties. We also apply the obtained results to the study of the spectral unit ball in Mn(C n ).

Abstract. We show that the symmetrized bidisc2 cannot be exhausted by domains biholomorphic to convex domains.

Abstract. In the paper we show that the Lempert property (i.e. the equality between the Lempert function and the Carathéodory distance) holds in the tetrablock, a bounded hyperconvex domain which is not biholomorphic to a convex domain. The question whether such an equality holds was posed in [3].