Some specialization theorems for families of abelian varieties

Consider an algebraic family pi: A -> B of abelian varieties, defined over (Q) over bar. We shall be concerned with properties of the generic fiber of A which are preserved on restricting to some (or 'many') suitable special fibers. We shall focus on instances like torsion for values of a section, endomorphism rings, existence of generic and special isogenies, illustrating some known results and some applications. Another, more recent, issue which we shall briefly discuss concerns the existence of abelian varieties over (Q) over bar not isogenous to a Jacobian. We shall conclude with a few comments on other specialization issues.