Beyond Serre's "Trees" in two directions: $\mathrm{\Lambda }$--trees and products of trees

Serre in “Trees” laid down the fundamentals of the theory of groups acting on simplicial trees. In particular, Bass-Serre theory makes it possible to extract information about the structure of a group from its action on a simplicial tree. Serre’s original motivation was to understand the structure of certain algebraic groups whose Bruhat–Tits buildings are trees. In this survey we will discuss the following generalizations of ideas from “Trees”: the theory of isometric group actions on $\mathrm{\Lambda }$-trees and the theory of lattices in the product of trees where we describe in more detail results on arithmetic groups acting on a product of trees.