Let N=1 be a square-free integer such that the modular curve X*0(N) has genus =2. We prove that X*0(N) is bielliptic exactly for 19 values of N, and we determine the automorphism group of these bielliptic curves. In particular, we obtain the first examples of nontrivial Aut(X*0(N)) when the genus of X*0(N) is =3. Moreover, we prove that the set of all quadratic points over Q for the modular curve X*0(N) with genus =2 and N square-free is not finite exactly for 51 values of N.

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