This article, written by Carlo Rovelli, provides a comprehensive overview of loop quantum gravity (LQG), a theoretical framework aiming to reconcile general relativity with quantum mechanics. LQG has been developing for about 25 years, with its origins in discussions at a 1986 workshop on quantum gravity and the first conference talk on a "loop-space representation of quantum general relativity" in 1987. The theory has evolved significantly, with contributions from numerous researchers, and has generated both enthusiasm and skepticism. Rovelli outlines the core ideas of LQG, emphasizing that it describes quantum states of geometry using functions of group elements associated with a graph. These states are invariant under local SU(2) gauge transformations, and the theory incorporates quantum discreteness of space, with areas and volumes quantized. The Penrose spin-geometry theorem plays a key role, showing that the algebraic structure of momentum operators in the Hilbert space defines a quantum geometry. This leads to the concept of "spin network states," which form an orthonormal basis for quantum states of space. The article also discusses transition amplitudes, which are defined using a two-complex (a combinatorial structure of faces, edges, and vertices) and resemble Feynman diagrams in quantum field theory. These amplitudes are interpreted as a discretization of the path integral for general relativity, incorporating the discrete nature of quantum geometry. Rovelli highlights the ultraviolet finiteness of the theory, attributed to the area gap in quantum geometry. The article addresses the problems LQG aims to solve: constructing a consistent quantum field theory whose classical limit is general relativity, addressing ultraviolet divergences, and developing a general covariant quantum field theory. It also contrasts LQG with other approaches like string theory and emphasizes that it does not aim to unify all forces or resolve the interpretation of quantum mechanics. Rovelli reflects on the historical development of LQG, including key ideas like the sum over geometries, Penrose's spin geometry theorem, Regge calculus, and the Wheeler-deWitt equation. He notes that the loop representation, based on Wilson loops and the Ashtekar variables, was crucial in the early development of LQG. The theory has evolved to focus on abstract graphs and spin networks, with nodes representing quanta of volume. Overall, Rovelli acknowledges the successes of LQG, such as its internal consistency and agreement with low-energy physics, while also recognizing its limitations and the open questions that remain. He concludes that while LQG may be incorrect as a description of quantum spacetime, it is also a promising approach that could be substantially correct.This article, written by Carlo Rovelli, provides a comprehensive overview of loop quantum gravity (LQG), a theoretical framework aiming to reconcile general relativity with quantum mechanics. LQG has been developing for about 25 years, with its origins in discussions at a 1986 workshop on quantum gravity and the first conference talk on a "loop-space representation of quantum general relativity" in 1987. The theory has evolved significantly, with contributions from numerous researchers, and has generated both enthusiasm and skepticism. Rovelli outlines the core ideas of LQG, emphasizing that it describes quantum states of geometry using functions of group elements associated with a graph. These states are invariant under local SU(2) gauge transformations, and the theory incorporates quantum discreteness of space, with areas and volumes quantized. The Penrose spin-geometry theorem plays a key role, showing that the algebraic structure of momentum operators in the Hilbert space defines a quantum geometry. This leads to the concept of "spin network states," which form an orthonormal basis for quantum states of space. The article also discusses transition amplitudes, which are defined using a two-complex (a combinatorial structure of faces, edges, and vertices) and resemble Feynman diagrams in quantum field theory. These amplitudes are interpreted as a discretization of the path integral for general relativity, incorporating the discrete nature of quantum geometry. Rovelli highlights the ultraviolet finiteness of the theory, attributed to the area gap in quantum geometry. The article addresses the problems LQG aims to solve: constructing a consistent quantum field theory whose classical limit is general relativity, addressing ultraviolet divergences, and developing a general covariant quantum field theory. It also contrasts LQG with other approaches like string theory and emphasizes that it does not aim to unify all forces or resolve the interpretation of quantum mechanics. Rovelli reflects on the historical development of LQG, including key ideas like the sum over geometries, Penrose's spin geometry theorem, Regge calculus, and the Wheeler-deWitt equation. He notes that the loop representation, based on Wilson loops and the Ashtekar variables, was crucial in the early development of LQG. The theory has evolved to focus on abstract graphs and spin networks, with nodes representing quanta of volume. Overall, Rovelli acknowledges the successes of LQG, such as its internal consistency and agreement with low-energy physics, while also recognizing its limitations and the open questions that remain. He concludes that while LQG may be incorrect as a description of quantum spacetime, it is also a promising approach that could be substantially correct.