  In the first part of our paper we discuss linear 2nd order differential
equations in the complex domain, especially Heun class equations, that is, the
Heun equation and its confluent cases. The second part of our paper is devoted
to Painlev\'e I-VI equations. Our philosophy is to treat these families of
equations in a unified way. This philosophy works especially well for Heun
class equations. We discuss its classification into 5 supertypes, subdivided
into 10 types (not counting trivial cases). We also introduce in a unified way
deformed Heun class equations, which contain an additional nonlogarithmic
singularity. We show that there is a direct relationship between deformed Heun
class equations and all Painlev\'e equations. In particular, Painlev\'e
equations can be also divided into 5 supertypes, and subdivided into 10 types.
This relationship is not so easy to describe in a completely unified way,
because the choice of the ''time variable'' may depend on the type. We describe
unified treatments for several possible ''time variables''.
