We present near-infrared spectro-imaging of the young planetary nebula NGC 7027 between 2.10 and 2.20 mu m with high spatial (0.5^'') and high spectral (8.7 km s^-1) resolution. The observations, made using BEAR at the CFHT, reveal the detailed morphology and kinematics of the ionized nebula (in the Brgamma line and 16 other atomic lines) and the surrounding molecular envelope (in the 1-0 S(1) line of H₂). The observations show that the ionized gas forms an elongated ( ~ 6 arcsec x 12 arcsec , PA = 32degr ), limb-brightened shell with an expansion velocity of 19.5 km s^-1 along the line of sight. The shell is composed of numerous small condensations and has nearly parallel sides with flattened ends that are not well matched by a uniform ellipsoidal model. Low level Brgamma emission is detected at high red- and blue-shifted velocities (+/- 55 km s^-1) along a bipolar axis at PA = 60degr that deviates significantly from that of the main nebula. The H₂ emission is distributed at the periphery of the ionized gas, in a limb-brightened, bi-conical shell (~ 10arcsec x 13arcsec , PA = 28degr ) with enhanced emission at the equator and complex structure at the ends of the major axis. The H₂ emission traces the inner edge of the extended molecular envelope seen in CO, and its distribution and intensity are well matched by model predictions of a high-density (n_H ~ 10⁶-10⁵ cm^-3) photo-dissociation region. The kinematic structure of the H₂ emission reveals a remarkable series of lobes and openings in the molecular shell. These features are point symmetric about the center, and the most prominent pair aligns with the high velocity, bipolar emission seen in Brgamma. These observations demonstrate recent activity by collimated outflows in NGC 7027, with a multiple, bipolar geometry. The interaction of the outflows with the surrounding envelope has significantly affected the morphology of the developing nebula and its environment, and their presence in this well-studied archetype underscores the general importance of outflows in the early shaping history of planetary nebulae. Based on observations collected at the Canada-France-Hawaii Telescope, operated by the National Research Council of Canada, the Centre National de la Recherche Scientifique de France, and the University of Hawaii.