until recently, it was unclear if there was a natural way to construct (compactified) moduli spaces of fano varieties. one approach to solving this problem is the k-moduli conjecture, which predicts that k-polystable fano varieties of fixed dimension and volume are parametrized by a projective good moduli space. in this talk, i will survey recent progress on this conjecture and discuss a result with yuchen liu and chenyang xu proving the openness of k-stability (a step in constructing k-moduli spaces).