Inverse problems arise in a variety of applications: image processing, finance, mathematical biology, and more. Mathematic models for these applications may involve integral equations, partial differential equations, and dynamical systems, and solution schemes are formulated by applying algorithms that incorporate regularization techniques and/or statistical approaches. In most cases these solutions schemes involve the need to solve a large-scale ill-conditioned linear system that is corrupted by noise and other errors. In this talk we describe Krylov subspace-based regularization approaches to solve these linear systems that combine direct matrix factorization methods on small subproblems with iterative solvers. The methods are very efficient for large scale inverse problems, they have the advantage that various regularization approaches can be used, and they can also incorporate methods to automatically estimate regularization parameters.