Many linear systems arising in practice are governed by rank-structured matrices. Examples include PDEs, integral equations, Gaussian process regression, etc. In this talk, we describe our recent work on fast direct algorithms that exploit such structure. These methods are of particular interest due to their exceptional robustness and high capacity for information reuse. Our main technical achievement is a linear-complexity matrix factorization as a generalized LU decomposition. This factorization permits fast multiplication/inversion and furthermore supports rapid updating. We anticipate that such techniques will be game-changing in environments requiring the analysis of many right-hand sides or the solution of many closely related systems, such as in protein design or other inverse problems. Similar applications abound in computational statistics and data analysis.