Abstract. We consider questions related to the arithmetic of elliptic curves and quadratic number fields, two of the most basic objects in Algebraic Number Theory and Arithmetic Geometry. We shall survey classical questions of Gauss,Cohen and Lenstra, Goldfeld, and Mazur on the `expected' behavior of ideal class groups of quadratic fields, Mordell-Weil groups of elliptic curves,and Shafarevich-Tate groups of elliptic curves. The purpose of this lecture is to review these fundamental questions and to present recent results. For example, we shall consider the frequency of elliptic curves of rank zero, and the frequency of the absence(resp. presence) of p-torsion in class groups and Shafarevich-Tate groups.