We introduce some special operator classes and study in terms of Berezin symbols their properties. In particular, we give some characterizations of compact operators and Schatten-von Neumann class operators in terms of Berezin symbols. We also consider some classes of compact operators on a Hilbert space H, which are generalizations of the well known Schatten-von Neumann classes of compact operators. Namely, for any number p, 0 < p < infinity, and the sequence w := (w(n))(n >= 0) of complex numbers w(n), n > 0, we define the following classes of compact operators on H: