We present and discuss the case for full waveform pixel and image acquisition and processing to enable LiDAR sensors to penetrate and reconstruct 3D surface maps through obscuring media. To that end, we review work on signal propagation, on scanning and arrayed sensors, on signal processing strategies for independent pixels and employing spatial context, on reducing complexity and accelerating processing by sensor design, algorithmic changes, compressed sensing, and parallel processing. We report several experimental studies on LiDAR imaging through complex media, and how these can inform the automotive LiDAR scenario. We conclude with a discussion of future development and potential for full waveform LiDAR (FWL).