This review discusses models of inflation and their predictions for primordial non-Gaussianity in density perturbations, which are thought to be the origin of cosmic structures. Non-Gaussianity is a key observable for distinguishing between inflationary scenarios and is a primary target for future Cosmic Microwave Background (CMB) satellite missions. The review provides a detailed overview of non-Gaussianity from both theoretical and observational perspectives, including all necessary tools to compute non-Gaussianity at second order in perturbation theory. It discusses new inflationary models rooted in modern particle physics that predict significant non-Gaussianity. The review is aimed at both astrophysicists and particle physicists, with useful tables summarizing theoretical and observational results on non-Gaussianity. The inflationary paradigm is based on the idea that the early universe underwent a period of accelerated expansion, driven by the energy density of a scalar field, the inflaton. This period solves the flatness, horizon, and monopole problems of the standard Big Bang model. The inflaton field's slow rolling down its potential generates primordial density perturbations, which are the seeds for large-scale structure and CMB anisotropies. After inflation ends, the inflaton oscillates and decays, reheating the universe. Alternative models, such as the curvaton and inhomogeneous reheating scenarios, generate curvature perturbations from other fields, avoiding the need for slow-roll conditions. These models assume the inflaton's initial perturbations are negligible. The curvaton scenario, for example, generates adiabatic perturbations when the curvaton decays into radiation. The inhomogeneous reheating scenario generates adiabatic perturbations due to variations in the inflaton's decay rate. The generation of gravitational waves is a generic prediction of accelerated de Sitter expansion, regardless of the mechanism for generating cosmological perturbations. Gravitational waves, detectable via B-mode polarization in CMB anisotropies, may provide indirect evidence of inflation. The power spectra of curvature and gravitational wave fluctuations provide a link between theory and observation. Current data from WMAP and Planck constrain the tensor-to-scalar amplitude ratio, r, to be less than 1.28 (95%). Non-Gaussianity is characterized by higher-order correlation functions, such as the bispectrum, which is the lowest-order statistic to distinguish non-Gaussian from Gaussian perturbations. The non-linearity parameter f_NL is used to quantify non-Gaussianity. Current observational data from WMAP and Planck provide constraints on f_NL, with the WMAP data giving a tightest limit of -58 < f_NL < 134 (95%). Other statistical tools, such as Minkowski functionals and spherical Mexican-hat wavelets, also provide constraints on f_NL. The review also discusses alternative models of inflation, including topologicalThis review discusses models of inflation and their predictions for primordial non-Gaussianity in density perturbations, which are thought to be the origin of cosmic structures. Non-Gaussianity is a key observable for distinguishing between inflationary scenarios and is a primary target for future Cosmic Microwave Background (CMB) satellite missions. The review provides a detailed overview of non-Gaussianity from both theoretical and observational perspectives, including all necessary tools to compute non-Gaussianity at second order in perturbation theory. It discusses new inflationary models rooted in modern particle physics that predict significant non-Gaussianity. The review is aimed at both astrophysicists and particle physicists, with useful tables summarizing theoretical and observational results on non-Gaussianity. The inflationary paradigm is based on the idea that the early universe underwent a period of accelerated expansion, driven by the energy density of a scalar field, the inflaton. This period solves the flatness, horizon, and monopole problems of the standard Big Bang model. The inflaton field's slow rolling down its potential generates primordial density perturbations, which are the seeds for large-scale structure and CMB anisotropies. After inflation ends, the inflaton oscillates and decays, reheating the universe. Alternative models, such as the curvaton and inhomogeneous reheating scenarios, generate curvature perturbations from other fields, avoiding the need for slow-roll conditions. These models assume the inflaton's initial perturbations are negligible. The curvaton scenario, for example, generates adiabatic perturbations when the curvaton decays into radiation. The inhomogeneous reheating scenario generates adiabatic perturbations due to variations in the inflaton's decay rate. The generation of gravitational waves is a generic prediction of accelerated de Sitter expansion, regardless of the mechanism for generating cosmological perturbations. Gravitational waves, detectable via B-mode polarization in CMB anisotropies, may provide indirect evidence of inflation. The power spectra of curvature and gravitational wave fluctuations provide a link between theory and observation. Current data from WMAP and Planck constrain the tensor-to-scalar amplitude ratio, r, to be less than 1.28 (95%). Non-Gaussianity is characterized by higher-order correlation functions, such as the bispectrum, which is the lowest-order statistic to distinguish non-Gaussian from Gaussian perturbations. The non-linearity parameter f_NL is used to quantify non-Gaussianity. Current observational data from WMAP and Planck provide constraints on f_NL, with the WMAP data giving a tightest limit of -58 < f_NL < 134 (95%). Other statistical tools, such as Minkowski functionals and spherical Mexican-hat wavelets, also provide constraints on f_NL. The review also discusses alternative models of inflation, including topological