This series of lectures is an invitation to the study of the combinatorics of words and roots in infinite Coxeter groups. The combinatorial interplay between words and roots plays a fundamental role in the study of these groups. They are for instance at the heart of the proof by Brink and Howlett that Coxeter groups are automatic.

We begin with an introduction to Coxeter groups, with particular emphasis on the relationship between reduced words and roots illustrated with classic examples.

In a second part, our focus will be the "weak order", which plays an important role in the study of reduced words. In finite Coxeter groups, it is a lattice and an orientation of the 1-skeleton of the permutahedron that provides a nice framework to construct generalized associahedra (via N. Reading's Cambrian lattices). We will discuss also a conjecture of Matthew Dyer that proposes a generalization of the framework weak order/reduced words to infinite Coxeter groups via certain generalizations of inversion sets.

Finally, we will discuss the case of infinite Coxeter groups in which the combinatorics of words and roots is mostly uncharted territory. On the way we will present a new framework for studying words and root systems that involves limits of roots and tilings of their convex hull. This last part will be based on joint works with M. Dyer, J.P. Labbé and V. Ripoll.