The classical Noether-Lefschetz theorem says that any curve in a very general surface X in P^3  of degree d \geq 4 is a restriction of a surface in the ambient space, in particular the Picard number of X is 1 (a property is very general if it holds in the complement of countably many proper closed subvarieties).The Noether-Lefschetz locus S_d is the locus of the degree d \geq 4 surfaces in P^3 whose Picard number is greater than 1. I will discuss 
generalizations and applications to singular ambient spaces, and also  
in particular properties of  the components of maximal codimension in the Noether-Lefschetz locus. (Base on work with Bruzzo and Lopez)