Measurements have been made to determine all sixteen elements of the Mueller scattering matrix for two types of nonspherical particles. Rounded particles of ammonium sulfate and nearly cubic particles of sodium chloride in the 0.1-1.0-mum size range have been prepared by nebulizing salt water solutions and drying the droplets. Scanning electron micrographs are used to determine size distributions used in Mie calculations of all matrix elements. The expected symmetry of the scattering matrices across the diagonal was confirmed, and the expected eight of the sixteen elements were found to be zero within measurement accuracy. The rounded particles were found accurately to obey Mie theory, while the cubic particles were poorly described by Mie theory for some matrix elements and some angles. Total intensity and linear polarization measurements are presented also for a series of increasing sizes of rounded and cubic particles. A discussion of the effect of nonsphericity on the various matrix elements is given, and applications of these results are given to analysis of particle properties in the laboratory, the clouds of Venus, reflection nebulae, the zodiacal light, and atmospheric particulates.