Abstract. 
This talk will be an introduction to the rotation theory for higher
dimensional torus mappings, which natually grew from the very classical
theory of rotation numbers for circle homeomorphisms. The general rotation
theory studies the dynamics of continuous torus mappings homotopic to the
identity. We will first define the notion of rotation sets, which is a
generalization of rotation numbers, for those mappings, and discuss some
general properties. After that we will focus on two special but
interesting cases: (1) (noninvertible) circle maps of degree 1,
(2) homeomorphisms of 2-torus. In both cases we will show that having
a "big" rotation set implies chaos in the dynamics of the torus mapping.