© 2019 American Mathematical Society. Let N ≥ 1 be a square-free integer such that the modular curve X0* (N) has genus ≥ 2. We prove that X0* (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 51 values of N.