On quadratic points of classical modular curves

The infinitude of the set of quadratic points of a non-singular projective curve C defined over a number field k, (that is, the points on C defined over a quadratic extension of k), is related with the geometric property that C is an hyperelliptic or a bielliptic curve, and such geometric property may be tested, for example, if one knows the automorphism group of C. In this survey paper we present the state of the art on finiteness or not for the set of quadratic points of classical modular curves. Moreover, we fix inaccuracies of the existing literature concerning automorphism groups of modular curves, and, we clarify the importance of k-points in the arithmetic statement between the relation of the curve C having an infinite set of quadratic points with C being an hyperelliptic or a bielliptic curve.