This paper presents a class of fast and accurate randomized algorithms for solving general linear systems and eigenvalue problems. These algorithms use fast randomized dimension reduction ("sketching") to accelerate standard subspace projection methods like GMRES and Rayleigh–Ritz. This approach allows the use of nontraditional bases for approximation subspaces, which are easier to construct. When the basis is numerically full rank, the new algorithms achieve accuracy similar to classic methods but run faster and may use less storage. Numerical experiments show significant advantages over optimized MATLAB routines, including a 70× speedup over GMRES and a 10× speedup over EIGS. The key idea is to combine subspace projection methods with randomized dimension reduction. This allows for faster solutions by incorporating easier-to-construct approximation subspaces. The algorithms are particularly effective for nonsymmetric linear systems and eigenvalue problems, and they can also be competitive for symmetric problems. The paper discusses the application of sketching to least-squares problems, linear systems via sketched GMRES, and eigenvalue problems via sketched Rayleigh–Ritz. It also addresses the construction of Krylov subspaces and the use of fast basis constructions. The algorithms are shown to be robust and efficient, with significant performance improvements in both theory and practice. The paper concludes with a discussion of related work and future research directions.This paper presents a class of fast and accurate randomized algorithms for solving general linear systems and eigenvalue problems. These algorithms use fast randomized dimension reduction ("sketching") to accelerate standard subspace projection methods like GMRES and Rayleigh–Ritz. This approach allows the use of nontraditional bases for approximation subspaces, which are easier to construct. When the basis is numerically full rank, the new algorithms achieve accuracy similar to classic methods but run faster and may use less storage. Numerical experiments show significant advantages over optimized MATLAB routines, including a 70× speedup over GMRES and a 10× speedup over EIGS. The key idea is to combine subspace projection methods with randomized dimension reduction. This allows for faster solutions by incorporating easier-to-construct approximation subspaces. The algorithms are particularly effective for nonsymmetric linear systems and eigenvalue problems, and they can also be competitive for symmetric problems. The paper discusses the application of sketching to least-squares problems, linear systems via sketched GMRES, and eigenvalue problems via sketched Rayleigh–Ritz. It also addresses the construction of Krylov subspaces and the use of fast basis constructions. The algorithms are shown to be robust and efficient, with significant performance improvements in both theory and practice. The paper concludes with a discussion of related work and future research directions.