This article introduces a new concept called "regularity structure," which provides an algebraic framework to describe functions and distributions through a local Taylor expansion around each point. The key innovation is to replace the classical polynomial model with purpose-built models tailored to the specific problem at hand. This allows for the description of both functions and large classes of distributions. The authors develop a calculus to perform operations such as multiplication, composition with smooth functions, and integration against singular kernels. This enables the formulation of fixed-point equations for a wide class of semilinear stochastic partial differential equations (SPDEs) driven by singular (typically random) inputs. The theory includes convergence results that allow these solutions to be interpreted as limits of classical solutions to regularized problems, possibly modified by diverging counterterms. These counterterms arise naturally through a "renormalization group" canonically defined in terms of the regularity structure associated with the given class of SPDEs. The theory also recovers many existing results on singular SPDEs, such as the KPZ equation, stochastic quantization equations, and Burgers-type equations, and shows that local solutions are actually "smooth" in the sense that they can be approximated locally by linear combinations of a fixed family of random functions or distributions. An example of a novel application is the construction of a natural Markov process that is symmetric with respect to the measure describing the $\Phi_3^4$ Euclidean quantum field theory. This process is conjectured to describe the Glauber dynamics of 3-dimensional ferromagnets near their critical temperature. The article covers the following main topics: 1. **Introduction**: Motivation, examples of interesting SPDEs, and the main problem. 2. **Abstract Regularity Structures**: Definitions, properties, and basic operations. 3. **Modelled Distributions**: Convergence criteria, reconstruction theorems, and symmetries. 4. **Multiplication**: Classical and composition with smooth functions. 5. **Integration Against Singular Kernels**: Extension theorem, multi-level Schauder estimates, and differentiation. 6. **Solutions to Semilinear SPDEs**: Fixed-point maps and renormalization procedures. 7. **Concrete Applications**: Convergence results for the parabolic Anderson model and the dynamical $\Phi_3^4$ model. The theory is designed to handle a broad class of locally subcritical SPDEs, providing a robust framework for their analysis and solution.This article introduces a new concept called "regularity structure," which provides an algebraic framework to describe functions and distributions through a local Taylor expansion around each point. The key innovation is to replace the classical polynomial model with purpose-built models tailored to the specific problem at hand. This allows for the description of both functions and large classes of distributions. The authors develop a calculus to perform operations such as multiplication, composition with smooth functions, and integration against singular kernels. This enables the formulation of fixed-point equations for a wide class of semilinear stochastic partial differential equations (SPDEs) driven by singular (typically random) inputs. The theory includes convergence results that allow these solutions to be interpreted as limits of classical solutions to regularized problems, possibly modified by diverging counterterms. These counterterms arise naturally through a "renormalization group" canonically defined in terms of the regularity structure associated with the given class of SPDEs. The theory also recovers many existing results on singular SPDEs, such as the KPZ equation, stochastic quantization equations, and Burgers-type equations, and shows that local solutions are actually "smooth" in the sense that they can be approximated locally by linear combinations of a fixed family of random functions or distributions. An example of a novel application is the construction of a natural Markov process that is symmetric with respect to the measure describing the $\Phi_3^4$ Euclidean quantum field theory. This process is conjectured to describe the Glauber dynamics of 3-dimensional ferromagnets near their critical temperature. The article covers the following main topics: 1. **Introduction**: Motivation, examples of interesting SPDEs, and the main problem. 2. **Abstract Regularity Structures**: Definitions, properties, and basic operations. 3. **Modelled Distributions**: Convergence criteria, reconstruction theorems, and symmetries. 4. **Multiplication**: Classical and composition with smooth functions. 5. **Integration Against Singular Kernels**: Extension theorem, multi-level Schauder estimates, and differentiation. 6. **Solutions to Semilinear SPDEs**: Fixed-point maps and renormalization procedures. 7. **Concrete Applications**: Convergence results for the parabolic Anderson model and the dynamical $\Phi_3^4$ model. The theory is designed to handle a broad class of locally subcritical SPDEs, providing a robust framework for their analysis and solution.