The (approximate) uniform sampling of graphs with a given degree sequence is an open problem that finds applications in, e.g., the analysis of social networks. The goal is to design an algorithm that, given a degree sequence, outputs every possible graphical realization of that degree sequence with approximately the same probability. In this talk I will provide some context for the problem, discuss some approaches to tackle it (with a focus on simple Markov chain Monte Carlo (MCMC) algorithms), and mention some generalizations/variations of the problem that I have recently worked on.