Some natural generalizations of sub-Hardy (de Branges-Rovnyak) spaces are Hilbert spaces of analytic functions in the disc, where the backward shift acts as a contraction. The sub-Bergman spaces introduced by K. Zhou are a different generalization which is interesting in its own right. These are essentially a particular case of Hilbert spaces of analytic functions in the disc, where the forward shift satisfies the famous hereditary inequality of S. Shimorin. The basic observation and the starting point of the talk is that all spaces considered here are reproducing kernel Hilbert spaces whose kernel is obtained by dividing a given kernel (like the Szego or Bergman kernel) by a normalized complete Nevanlinna-Pick kernel. The aim is to deduce some general properties of these objects. We derive a useful formula for the norm and discuss some approximation results as well as some embedding theorems. This is a report about recent joint work with F. Weistrom Dahlin as well as previous work joint with B. Malman.