  The warped product $N_1\times_f N_2$ of two Riemannian manifolds $(N_1,g_1)$
and $(N_2,g_2)$ is the product manifold $N_1\times N_2$ equipped with the
warped product metric $g=g_1+f^2 g_2$, where $f$ is a positive function on
$N_1$. The notion of warped product manifolds is one of the most fruitful
generalizations of Riemannian products. Such notion plays very important roles
in differential geometry as well as in physics, especially in general
relativity. Warped product manifolds have been studied for a long period of
time. In contrast, the study of warped product submanifolds was only initiated
around the beginning of this century.
  In this article we survey important results on warped product submanifolds in
various ambient manifolds. It is the author's hope that this survey article
will provide a good introduction on the theory of warped product submanifolds
as well as a useful reference for further research on this vibrant research
subject.
