The Yang-Baxter equation is a fundamental equation appearing in many different contexts of physics. In 1992, Drinfeld proposed a mathematical and combinatorial approach to the problem of classifying solutions, which was followed by a seminal article of Etingof, Schedler and Soloviev in 1999. Throughout this talk, I will present the basic notions behind the study of set-theoretical solutions to the Yang-Baxter equation, mainly from a group theoretical and Garside perspective. In this sense, I will focus on the known parallels between groups associated to solutions and the classical properties of finite Coxeter groups. In the last part, I'll talk about a new construction of Hecke algebras, inspired by the theory of Iwahori-Hecke algebras for Coxeter groups.