ABSTRACT: In his book `L'analysis situs et la g\'eom\'etrie alg\'ebrique', S. Lefschetz proved two fundamental results on the topology of algebraic varieties: the Hyperplane Section Theorem and so-called `Second Lefschetz Theorem'. These results give a comparison between the homology groups of a non-singular irreducible projective variety X in {CP}^n
and the homology groups of a generic hyperplane section of X. Precisely, the Hyperplane Section Theorem says that, for a generic hyperplane L, the natural map

H_q(L \cap X) \rightarrow H_q(X)
 
is an isomorphism for q\leq \dim X-2 and an epimorphism for q = \dim X-1. The Second Lefschetz Theorem describes the kernel of the map

H_{\dim X-1}(L \cap X) \rightarrow H_{\dim X-1}(X)

in terms of `vanishing cycles' that appear in a generic pencil of hyperplanes.

In this talk, I will discuss generalizations of these theorems to quasi-projective varieties and homotopy groups.