Generally speaking, specialization theorems assert that a property of a family of objects which depend on parameters, remains valid for suitable sets of values of the parameters. A typical instance is Bertini irreducibility theorem in algebraic geometry.

In this talk, after giving a simple example which shows limitations of such principle, we shall survey on comparatively recent instances, especially of arithmetical nature, and concerning families of elliptic curves and abelian varieties. We shall stick to basic examples while illustrating general theorems.