  In this paper we study the Lorenz equations using the perspective of the
Conley index theory. More specifically, we examine the evolution of the strange
set that these equations posses throughout the different values of the
parameter. We also analyze some natural Morse decompositions of the global
attractor of the system and the role of the strange set in these
decompositions. We calculate the corresponding Morse equations and study their
change along the successive bifurcations. In addition, we formulate and prove
some theorems which are applicable in more general situations. These theorems
refer to Poincar\'{e}-Andronov-Hopf bifurcations of arbitrary codimension,
bifurcations with two homoclinic loops and a study of the role of the
travelling repellers in the transformation of repeller-attractor pairs into
attractor-repeller ones.
