BPS counting in string compactifications

I'll review the known results about BPS indices, which encode in particular the entropy of BPS black holes, appearing in string compactifications down to four dimensions with various number of supersymmetries. First, I'll recall the well-known results about BPS states in $\mathcal{N}=8$ and $\mathcal{N}=4$ compactifications, and then present what is known about them in the $\mathcal{N}=2$ case. Depending on time, I hope to cover some recent advances where an important role was played by (mock) modular symmetry.