Let N ≥ 1 be a square-free integer such that the modular curve X0*(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(X0*(N)) when the genus of X0*(N) is ≥ 3. Moreover, we prove that the set of all quadratic points over ℚ for the modular curve X0*(N) with genus ≥ 2 and N square-free is not finite exactly for 50 values of N.