Despite occasional experimental hints, medium-range structural order in covalently bonded amorphous semiconductors had largely escaped detection until the advent of fluctuation electron microscopy (FEM) in 1996. Using FEM, we find that every sample of amorphous silicon and germanium we have investigated, regardless of deposition method or hydrogen content, is rich in medium-range order. The paracrystalline structural model, which consists of small, topologically ordered grains in an amorphous matrix, is consistent with the FEM data, and is rendered diffraction amorphous by strain effects. We present measurements on hydrogenated amorphous silicon deposited by different methods, some of which are reported to have greater stability against the Staebler-Wronski effect. The matrix material of these samples is relatively similar, but the order changes in different ways upon both light soaking and thermal annealing. Some materials are inhomogeneous, with either nanocrystalline inclusions or large area-to-area variation in the medium-range order. We discuss the implications of and future directions for understanding medium-range order. (C) 2002 Elsevier Science B.V. All rights reserved.