  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 (we call these ``generalized Lagrangian
submanifolds'' in our paper), we introduce and study three other classes of
submanifolds. 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.
  We then support our ``symplectic/Lagrangian viewpoint'' on the submanifolds
introduced by Gualtieri and Hitchin by defining the ``generalized complex
category'', modelled on the constructions of Guillemin-Sternberg and Weinstein.
We argue that our approach may be useful for the quantization of generalized
complex manifolds.
