The congruent number problem is to find a right triangle with rational sides whose area is a given positive integer. This problem can be traced back at least to 10-th century in Arab manuscripts. It turns out to be a beautiful example of the rich modern theory of the arithmetic of elliptic curves, or it is more accurate to say that this theory grew out of the study of this problem. In the two lectures, we survey arithmetic theory of elliptic curves and introduce some progress on the congruent number problem.