The Thesis: Radar/sonar/lidar are very well-studied means for the detection of objects that are distant from the observer and for the estimation of such physical properties of the object as its distance from the observer, surface reflectivity, rotation, and velocity. This dissertation focuses instead on the novel issue of estimating the surface of a three-dimensional convex room that is empty except for the acoustic-based measurement system introduced by the observer. While acoustic sounding is the essence of sonar, our objective of characterizing a convex polyhedral room is unlike any exploration undertaken hitherto. Approach and Results: We first deploy a single omnidirectional (its gain pattern need not be directionally uniform but is always positive) sound source at a location of our choosing that will be taken to be the origin of our coordinate system. This source will be controlled to emit a short pulse of duration T * and known signal shape s(t). Subsequent pulses may be generated provided the interpulse interval is sufficiently long. We then deploy an array of omnidirectional (again, they can have directiondependent gains that are always positive) microphones at known locations. We assume that the walls of the room are not only planar but also have surfaces that yield specular acoustic reflection that is akin to the reflection of light from a mirror. Each microphone is monitored to record both the direct line-of-sight pulse from the source and a first received echo. The records of the originating pulse and the two pulses recorded at each microphone are then processed centrally to infer the placement of the walls of the room. As we do not know the true number W of walls, we cannot guarantee that we will detect each of them. As we shall see, it takes four microphones receiving a first echo from the same wall to enable us to locate that wall. Hence, we would need a minimum of 4W microphones to ensure detection of all walls. As W is unknown, the detection of all walls cannot be guaranteed. Hence, our approach is only known to generate a convex polyhedral upper bound to the true convex polyhedral room. We will treat both the case of noiseless reception at the microphones and the more complex but more realistic case of noisy reception. The theoretical results achieved are sometimes supplemented by simulated examples and/or by a laboratory experiment.