Study of equivalences and symmetries of real submanifolds in
complex space goes back to the classical work of Poincar\'e  and Cartan
and was deeply developed in later work of Tanaka and Chern and Moser. This
work initiated far going research in the area (since 1970's till present),
which is dedicated to questions of regularity of mappings between real
submanifolds in complex space, unique jet determination of mappings,
solution of the equivalence problem, and study of automorphism groups of
real submanifolds.

Current state of the art and methods involved provide satisfactory (and
sometimes complete) solution for the above mentioned problems in
nondegenerate settings. However, very little is known for more degenerate
situations, i.e., when real submanifolds under consideration admit certain
degeneracies of the CR-structure.

The recent CR (Cauchey-Riemann Manifolds) - DS (Dynamical Systems)
technique, developed in our joint work with Shafikov and Lamel, suggests
to replace a real submanifold with a CR-singularity by an appropriate
dynamical systems. This technique has recently enabled us to solve a
number of long-standing problems in CR-geometry, in particular, related to
a Conjecture by Treves and that by Ebenfelt and Huang.

The technique also has applications to Dynamics.

In this talk, we give an overview of the technique and the results
obtained recently by using it.