These lecture notes provide an introduction to the analysis of the infinity Laplacian. This topic has undergone significant development in recent years within the field of nonlinear partial differential equations. The material is based on minicourses taught online during the pandemic at the Universidade Federal da Paraíba (João Pessoa, Brazil), and in person at the 24th Brazilian Mathematical Colloquium held at IMPA (Rio de Janeiro, Brazil) in July 2023. Shorter versions of these lectures were previously delivered by Miguel Urbano at the Universidade Federal do Ceará (Fortaleza, Brazil) during the Summer of 2013, at Aalto University (Helsinki, Finland) in the Spring of 2013, and at KAUST (Thuwal, Saudi Arabia) in early 2017. The first author also presented a condensed version at the Universidade Federal do Rio de Janeiro (Rio de Janeiro, Brazil) during the Summer of 2020.