"The classical uniformization theorem for Riemann surfaces is well-established for both compact and non-compact connected surfaces. However, in the discrete setting, the discrete uniformization theorem is only known for compact surfaces. In this talk, we will propose a discrete uniformization problem for all surfaces. We will discuss our recent work on the uniformization problem for all polyhedral surfaces of non-abelian fundamental groups and the discrete Schwarz-Pick-Ahlfors lemma. This is joint work with Yanwen Luo."