  This survey is dedicated to the 100th anniversary of Mark Iosifovich Vishik
and is based on a number of mini-courses taught by the author at University of
Surrey (UK) and Lanzhou University (China). It discusses the classical and
modern results of the theory of attractors for dissipative PDEs including
attractors for autonomous and non-autonomous equations, dynamical systems in
general topological spaces, various types of trajectory, pullback and random
attractors, exponential attractors, determining functionals and inertial
manifolds as well as the dimension theory for the above mentioned classes of
attractors. The theoretical results are illustrated by a number of clarifying
examples and counter-examples.
