Non-line-of-sight (NLOS) imaging is the inference of the properties of objects or scenes outside of the direct line-of-sight of the observer. Such inferences can range from a 2D photograph-like image of a hidden area, to determining the position, motion or number of hidden objects, to 3D reconstructions of a hidden volume. NLOS imaging has many enticing potential applications, such as leveraging the existing hardware in many automobiles to identify hidden pedestrians, vehicles or other hazards and hence plan safer trajectories. Other potential application areas include improving navigation for robots or drones by anticipating occluded hazards, peering past obstructions in medical settings, or in surveying unreachable areas in search-and-rescue operations. Most modern NLOS imaging methods fall into one of two categories: active imaging methods that have some control of the illumination of the hidden area, and passive methods that simply measure light that already exists. This thesis introduces two NLOS imaging methods, one of each category, along with modeling and data processing techniques that are more broadly applicable. The methods are linked by their use of objects (‘occluders’) that reside somewhere between the observer and the hidden scene and block some possible light paths. Computational periscopy, a passive method, can recover the unknown position of an occluding object in the hidden area and then recover an image of the hidden scene behind it. It does so using only a single photograph of a blank relay wall taken by an ordinary digital camera. We develop also a framework using an optimized preconditioning matrix to improve the speed at which these reconstructions can be made and greatly improve the robustness to ambient light. Lastly, we develop tools necessary to demonstrate recovery of scenes at multiple unknown depths – paving the way towards three-dimensional reconstructions. Edge-resolved transient imaging, an active method, enables the formation of 2.5D representations – a plan view plus heights – of large-scale scenes. A pulsed laser illuminates spots along a small semi-circle on the floor, centered on the edge of a vertical wall such as in a doorway. The wall edge occludes some light paths, only allowing the laser light reflecting off of the floor to illuminate certain portions of the hidden area beyond the wall, depending on where along the semi-circle it is illuminating. The time at which photons return following a laser pulse is recorded. The occluding wall edge provides angular resolution, and time-resolved sensing provides radial resolution. This novel acquisition strategy, along with a scene response model and reconstruction algorithm, allow for 180° field of view reconstructions of large-scale scenes unlike other active imaging methods. Lastly, we introduce a sparsity penalty named mutually exclusive group sparsity (MEGS), that can be used as a constraint or regularization in optimization problems to promote solutions in which certain components are mutually exclusive. We explore how this penalty relates to other similar penalties, develop fast algorithms to solve MEGS-regularized problems, and demonstrate how enforcing mutual exclusivity structure can provide great utility in NLOS imaging problems.