  This is a self-contained tour of the Conley index and connection matrices.
The starting point is Conley's fundamental theorem of dynamical systems. There
is a short stop at the necessary topological background, before we proceed to
the basic properties of the index. Then, the itinerary passes through the
construction of connection matrices with a panoramic view of the applications:
detect heteroclinic orbits arising in delay differential equations, and partial
differential equations of parabolic type. The ride will be filled with examples
and figures.
