  Let $\gg$ be a complex reductive Lie algebra and $\kk\subset\gg$ be any
reductive in $\gg$ subalgebra. We call a $(\gg,\kk)$-module $M$ bounded if the
$\kk$-multiplicities of $M$ are uniformly bounded. In this paper we initiate a
general study of simple bounded $(\gg,\kk)$-modules. We prove a strong
necessary condition for a subalgebra $\kk$ to be bounded (Corollary
\ref{cor1.6}), i.e. to admit an infinite-dimensional simple bounded
$(\gg,\kk)$-module, and then establish a sufficient condition for a subalgebra
$\kk$ to be bounded (Theorem \ref{thGroups2}). As a result we are able to
classify all maximal bounded reductive subalgebras of $\gg=\sl(n)$.
  In the second half of the paper we describe in detail simple bounded
infinite-dimensional $(\gg,\sl(2))$-modules, and in particular compute their
characters and minimal $\sl(2)$-types. We show that if $\sl(2)$ is a bounded
subalgebra of $\gg$ which is not contained in a proper ideal of $\gg$, then
$\gg\simeq \sl(2)\oplus \sl(2), \sl(3),\sp(4)$; alltogether, up to conjugation
there are five possible embeddings of $\sl(2)$ as a bounded subalgebra into
$\gg$ as above. In two of these cases $\sl(2)$ is a symmetric subalgebra, and
many results about simple bounded $(\gg,\sl(2))$-modules are known. A case
where our results are entirely new is the case of a principal
$\sl(2)$-subalgebra in $\sp(4)$.
