Fano varieties are in some sense the simplest type of algebraic varieties. They are the algebraic analogue of manifolds with positive curvature, such as spheres. In low dimensions one can classify Fano varieties (where for an algebraic geometer low means up to dimension three) and as the dimension increases, they form bounded families, so that one can in principle classify Fano varieties in all dimensions. In this talk we explain some of the known and conjectured results, both for an explicit classification, and for some of the boundedness results.