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

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