Gorenstein rings are very common and significant in many areas of mathematics. The following are two important and widely open problems in commutative algebra and algebraic geometry:

1. characterization of possible Hilbert functions of Gorenstein K-algebras; and
2. classification of Gorenstein K-algebras with given Hilbert functions.

Recently, in a joint work with M.E. Rossi, we obtained partial results to these problems in some cases (K-algebras of socle degree 4). In this talk, we will discuss these new developments.