Multi-Agent Path Finding (MAPF) is the challenging problem of computing collision-free paths for multiple agents. Algorithms for solving MAPF can be categorized on a spectrum. At one end are (bounded-sub)optimal algorithms that can find high-quality solutions for small problems. At the other end are unbounded-suboptimal algorithms that can solve large problems but usually find low-quality solutions. In this paper, we consider a third approach that combines the best of both worlds: anytime algorithms that quickly find an initial solution using efficient MAPF algorithms from the literature, even for large problems, and that subsequently improve the solution quality to near-optimal as time progresses by replanning subgroups of agents using Large Neighborhood Search. We compare our algorithm MAPF-LNS against a range of existing work and report significant gains in scalability, runtime to the initial solution, and speed of improving the solution.