The optical emission lines of the Cygnus Loop, IC 443, Vela X, Puppis A, LMC N49, and LMC N63A were observed at very high spatial and spectral resolution with 4-m echelle spectrographs. Results indicate that the velocity dispersions of knots in the filaments of the nearby galactic supernova remnants (SNRs) result primarily from turbulence within the emitting regions. The line-of-sight velocity dispersions (half-widths at half maximum) within the observed knots are on the order of 10-30 km s('-1), including associated thermal velocity dispersions of roughly 15 km s('-1) for hydrogen, 6 km s('-1) for oxygen, and 4 km s('-1) for nitrogen. In general, the knots move randomly relative to each other with speeds of 10-30 km s('-1), although relative speeds as large as 80 km s('-1) are observed in Puppis A. Occasionally, diffuse components of the filaments are observed. These diffuse components have velocity dispersions between 55 and 85 km s('-1), but the relative contributions of turbulent and thermal motion to the dispersions are unknown. The galactic observations are interpreted in terms of the "microscopic" structure of SNR filaments and knots. Velocity dispersions between 5 and 140 km s('-1) are observed in the distant SNRs of the Large Magellanic Cloud (LMC). Velocity dispersions of 5-20 km s('-1) are associated with spatially extended areas that possess the same radial velocities as the LMC. The associated thermal velocity dispersions for these areas are approximately 13 km s('-1) for hydrogen, 3.5 km s('-1) for nitrogen, 3.3 km s('-1) oxygen, and 2.1 km s('-1) for sulfur. Larger velocity dispersions of 30-140 km s('-1) are associated with smaller areas that are moving relative to the spatially extended areas. These relative velocities may be as large as 250 km s('-1). The contributions of thermal and bulk motion to the velocity dispersions of the smaller areas cannot be determined with kinematic methods. The LMC observations are interpreted in the context of SNR evolution in cloudy interstellar media.