Many imaging and compressed-sensing applications seek to approximate a signal as a linear combination of only a few elementary atoms drawn from a large collection. This is known as sparse reconstruction. The basis pursuit (BP) approach minimizes the 1-norm of the solution, and the BP denoising (BPDN) approach balances it against the least-squares fit. I will discuss the role of duality in revealing some unexpected and useful properties of these problems, and will show how they can lead to practical, large-scale algorithms.