This paper introduces a new mathematical framework called "regularity structures" to analyze and solve a wide class of semilinear stochastic partial differential equations (SPDEs). The theory provides an algebraic structure that allows for the description of functions and distributions through a local Taylor expansion, enabling the handling of singular inputs like white noise. The main idea is to replace classical polynomial models with purpose-built models tailored to the specific problem, allowing for the description of both functions and distributions. The theory includes a calculus for operations such as multiplication, composition with smooth functions, and integration against singular kernels, necessary for solving SPDEs. This framework allows for the first time a mathematically rigorous meaning to many interesting stochastic PDEs in physics. The theory also includes convergence results that interpret solutions as limits of classical solutions to regularized problems, possibly modified by diverging counterterms arising from a canonical renormalization group. The theory recovers many existing results on singular stochastic PDEs, such as the KPZ equation, stochastic quantization equations, and Burgers-type equations, and presents them as particular instances of a unified framework. A key insight is that local solutions are "smooth" in the sense that they can be approximated locally to arbitrarily high degree as linear combinations of a fixed family of random functions or distributions. The paper also presents a novel application: solving the long-standing problem of building a natural Markov process symmetric with respect to the finite volume measure of Euclidean quantum field theory. It is conjectured that this process describes the Glauber dynamics of 3-dimensional ferromagnets near their critical temperature. The theory is applied to various examples, including the stochastic quantization of the $\Phi^4$ quantum field theory, the continuous parabolic Anderson model, and the dynamical $\Phi_3^4$ model. The results show convergence of solutions to a limit, with the convergence understood in terms of probability on spaces of continuous trajectories. The analysis involves renormalization procedures, including the use of diverging counterterms and the interpretation of solutions as limits of regularized equations. The theory also incorporates symmetries and renormalization group structures, providing a robust framework for analyzing a wide range of SPDEs.This paper introduces a new mathematical framework called "regularity structures" to analyze and solve a wide class of semilinear stochastic partial differential equations (SPDEs). The theory provides an algebraic structure that allows for the description of functions and distributions through a local Taylor expansion, enabling the handling of singular inputs like white noise. The main idea is to replace classical polynomial models with purpose-built models tailored to the specific problem, allowing for the description of both functions and distributions. The theory includes a calculus for operations such as multiplication, composition with smooth functions, and integration against singular kernels, necessary for solving SPDEs. This framework allows for the first time a mathematically rigorous meaning to many interesting stochastic PDEs in physics. The theory also includes convergence results that interpret solutions as limits of classical solutions to regularized problems, possibly modified by diverging counterterms arising from a canonical renormalization group. The theory recovers many existing results on singular stochastic PDEs, such as the KPZ equation, stochastic quantization equations, and Burgers-type equations, and presents them as particular instances of a unified framework. A key insight is that local solutions are "smooth" in the sense that they can be approximated locally to arbitrarily high degree as linear combinations of a fixed family of random functions or distributions. The paper also presents a novel application: solving the long-standing problem of building a natural Markov process symmetric with respect to the finite volume measure of Euclidean quantum field theory. It is conjectured that this process describes the Glauber dynamics of 3-dimensional ferromagnets near their critical temperature. The theory is applied to various examples, including the stochastic quantization of the $\Phi^4$ quantum field theory, the continuous parabolic Anderson model, and the dynamical $\Phi_3^4$ model. The results show convergence of solutions to a limit, with the convergence understood in terms of probability on spaces of continuous trajectories. The analysis involves renormalization procedures, including the use of diverging counterterms and the interpretation of solutions as limits of regularized equations. The theory also incorporates symmetries and renormalization group structures, providing a robust framework for analyzing a wide range of SPDEs.