**Summary:** This text provides an introduction to the Gaussian free field (GFF) and Liouville quantum gravity (LQG), two central concepts in mathematical physics and probability theory. The GFF is a random distribution on a given surface, characterized by its covariance involving the Green function, which measures the expected time a Brownian motion spends near a point before exiting a domain. The book begins with a detailed treatment of the discrete GFF, including its definition, properties, and construction, followed by a discussion of the continuous Green function and its role in the continuum GFF. Key topics include the construction of the Liouville measure, which is a family of measures defined as $ \mathcal{M}^{\gamma}(\mathrm{d}x) = \exp(\gamma h(x)) \mathrm{d}x $, where $ h $ is the GFF and $ \gamma $ is a coupling constant. The book also explores Gaussian multiplicative chaos (GMC), a framework for constructing random measures from the GFF, and its connections to random surfaces and statistical mechanics. The text delves into the theory of Liouville quantum gravity and conformal field theory, emphasizing their deep connections and the role of the GFF in both. It discusses the conformal invariance of the GFF, the construction of quantum surfaces, and the relationship between the GFF and Schramm–Loewner Evolution (SLE). The book also covers the mating of trees, a powerful concept in LQG that relates to the structure of random planar maps. The authors provide a comprehensive treatment of the GFF with various boundary conditions, including Neumann and Dirichlet, and explore the implications of these conditions in the context of LQG. The text includes detailed proofs of key results, such as the Markov property of the GFF, the conformal invariance of the Green function, and the KPZ theorem, which relates the scaling dimensions of geometric quantities in LQG to those in critical statistical mechanics. The book is structured to guide readers from the basics of the GFF to advanced topics in LQG, with a focus on both theoretical foundations and applications. It includes exercises and references to further reading, making it a valuable resource for researchers and students in probability theory, mathematical physics, and related fields.**Summary:** This text provides an introduction to the Gaussian free field (GFF) and Liouville quantum gravity (LQG), two central concepts in mathematical physics and probability theory. The GFF is a random distribution on a given surface, characterized by its covariance involving the Green function, which measures the expected time a Brownian motion spends near a point before exiting a domain. The book begins with a detailed treatment of the discrete GFF, including its definition, properties, and construction, followed by a discussion of the continuous Green function and its role in the continuum GFF. Key topics include the construction of the Liouville measure, which is a family of measures defined as $ \mathcal{M}^{\gamma}(\mathrm{d}x) = \exp(\gamma h(x)) \mathrm{d}x $, where $ h $ is the GFF and $ \gamma $ is a coupling constant. The book also explores Gaussian multiplicative chaos (GMC), a framework for constructing random measures from the GFF, and its connections to random surfaces and statistical mechanics. The text delves into the theory of Liouville quantum gravity and conformal field theory, emphasizing their deep connections and the role of the GFF in both. It discusses the conformal invariance of the GFF, the construction of quantum surfaces, and the relationship between the GFF and Schramm–Loewner Evolution (SLE). The book also covers the mating of trees, a powerful concept in LQG that relates to the structure of random planar maps. The authors provide a comprehensive treatment of the GFF with various boundary conditions, including Neumann and Dirichlet, and explore the implications of these conditions in the context of LQG. The text includes detailed proofs of key results, such as the Markov property of the GFF, the conformal invariance of the Green function, and the KPZ theorem, which relates the scaling dimensions of geometric quantities in LQG to those in critical statistical mechanics. The book is structured to guide readers from the basics of the GFF to advanced topics in LQG, with a focus on both theoretical foundations and applications. It includes exercises and references to further reading, making it a valuable resource for researchers and students in probability theory, mathematical physics, and related fields.