Sheaths in low temperature collisionless and weakly collisional plasmas are often viewed as simple examples of nonlinear physics. How well do we understand them? Closer examination indicates that they are far from simple. Moreover, many predicted sheath properties have not been experimentally verified and even the appropriate "Bohm velocity" for often encountered two-ion species plasma is unknown. In addition, a variety of sheathlike structures, e.g., double layers, can exist, and many two-and three-dimensional sheath effects have not been considered. Experimental studies of sheaths and presheaths in weakly collisional plasmas are described. A key diagnostic is emissive probes operated in the "limit of zero emission." Emissive probes provide a sensitive diagnostic of plasma potential with a resolution approaching 0.1 V and a spatial resolution of 0.1 cm. Combined with planar Langmuir probes and laser-induced fluorescence, they have been used to investigate a wide variety of sheath, presheath, and sheathlike structures. Our experiments have provided some answers but have also raised more questions.