We first provide a brief review of work on finite-time singularity formation and well-posedness theory in fluid-interface problems. This work has involved both theoretical and computational approaches. The second part of the talk focuses on recent work on the water wave problem. The question of the well-posedness of the initial value problem for water waves has a celebrated history. It is known that the problem is well-posed, but the theory is subtle and the design of approximate models or convergent numerical schemes is likewise delicate. We demonstrate that a truncated system of equations for water waves that forms the basis of a widely-used numerical method is ill-posed. To show ill-posedness, we analytically extend the equations to complex values of the independent variable and analyze the initial value problem in the complex plane.