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These notes are devoted to lattices in products of trees and related topics.They provide an introduction to the construction, by M. Burger and S. Mozes, ofexamples of such lattices that are simple as abstract groups. Two features ofthat construction are emphasized: the relevance of non-discrete locally compactgroups, and the two-step strategy in the proof of simplicity, addressingseparately, and with completely different methods, the existence of finite andinfinite quotients. A brief history of the quest for finitely generated andfinitely presented infinite simple groups is also sketched. A comparison withMargulis' proof of Kneser's simplicity conjecture is discussed, and therelevance of the Classification of the Finite Simple Groups is pointed out. Afinal chapter is devoted to finite and infinite quotients of hyperbolic groupsand their relation to the asymptotic properties of the finite simple groups.Numerous open problems are discussed along the way.

Finite and infinite quotients of discrete and indiscrete groups