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 two points. This is joint work with Frederick Saia.