Direct images of SN 1987A taken during periods of good seeing at La Silla Observatory are combined with spectra to map the velocity and spatial structure of the dust and gas that surround SN 1987A. The data from 1989 December show that the supernova is embedded in a filamentary nebula of mixed gas and dust that is morphologically and kinematically similar to planetary nebulae. The narrowness of the filaments, their shape, their radial velocities, and the lack of detectable expansion in the plane of the sky imply that we are viewing a physical structure, not just the SN 1987A light curve seen reflected from a continuous sheet of background material. This inner, highly structured nebula is immersed in an outer, larger nebula which may be due to reflected light from Sk -69°202, the progenitor of SN 1987A. Alternatively, if it is due to reflected light from SN 1987A, it must lie in front of the supernova.