I will discuss formulations and algorithms for computing sparse optimal controls in systems governed by PDEs. These sparse solutions can guide the placement of control devices in applications.  After reviewing results for elliptic and parabolic PDEs, I will focus on recent work on sparse optimal control governed by linear PDEs with uncertain coefficients. Here, we aim at finding stochastic controls that share the same sparsity structure.  To solve the corresponding non-smooth and high-dimensional optimization problem, we propose variants of an iterative re-weighted least squares algorithm, which iterates over functions defined on the physical space only and thus avoids approximation in random space.  This is collaborative work with Chen Li.