Isogeny-based cryptography is a young and very active subfield of post-quantum cryptography based on the mathematical problem of constructing an isogeny, i.e. a well-behaved map, between two given elliptic curves over finite fields. Since 1997 and especially in the past 10-15 years, several schemes based on isogenies have been proposed, mostly within the context of postquantum key exchange and post-quantum digital signatures. In this course, after reviewing the mathematical background (elliptic curves, isogenies, supersingular isogeny graphs, etc.), we will give a brief history of isogeny-based cryptography and describe some isogeny-based protocols. In particular we will focus our attention on the CGL hash function, which will give us the opportunity to address some important open problems related to the field.