The main goal of our paper is the study of several classes of
submanifolds of generalized complex manifolds. Along with the
generalized complex submanifolds defined by Gualtieri and Hitchin
in \cite{Gua}, \cite{H3} (we call these ``generalized Lagrangian
submanifolds'' in our paper), we introduce and study three other
classes of submanifolds and their relationships.
For generalized complex manifolds that
arise from complex (resp., symplectic) manifolds, all three
classes specialize to complex (resp., symplectic) submanifolds. In
general, however, all three classes are distinct. We discuss some
interesting features of our theory of submanifolds, and illustrate
them with a few nontrivial examples. Along the way, we obtain a
complete and explicit classification of all linear
generalized complex structures.
We then support our ``symplectic/Lagrangian viewpoint'' on the
submanifolds introduced in \cite{Gua}, \cite{H3} by defining the
``generalized complex category'', modelled on the constructions of
Guillemin-Sternberg \cite{GS} and Weinstein \cite{Wei2}. We argue
that our approach may be useful for the quantization of
generalized complex manifolds.