In this talk, I explain how we work towards completely classifying all bielliptic Shimura curves X_0^D(N) with nontrivial level N, extending a result of Rotger that provided such a classification for level one. This allows us to determine the list of all pairs (D,N) for which X_0^D(N) has infinitely many degree 2 points, up to 3 explicit possible exceptions. This is joint work with Frederick Saia.