  Inspired by recent activities on Whittaker modules over various (Lie)
algebras we describe some general framework for the study of Lie algebra
modules locally finite over a subalgebra. As a special case we obtain a very
general setup for the study of Whittaker modules, which includes, in
particular, Lie algebras with triangular decomposition and simple Lie algebras
of Cartan type. We describe some basic properties of Whittaker modules,
including a block decomposition of the category of Whittaker modules and
certain properties of simple Whittaker modules under some rather mild
assumptions. We establish a connection between our general setup and the
general setup of Harish-Chandra subalgebras in the sense of Drozd, Futorny and
Ovsienko. For Lie algebras with triangular decomposition we construct a family
of simple Whittaker modules (roughly depending on the choice of a pair of
weights in the dual of the Cartan subalgebra), describe their annihilators and
formulate several classification conjectures. In particular, we construct some
new simple Whittaker modules for the Virasoro algebra. Finally, we construct a
series of simple Whittaker modules for the Lie algebra of derivations of the
polynomial algebra, and consider several finite dimensional examples, where we
study the category of Whittaker modules over solvable Lie algebras and their
relation to Koszul algebras.
