The Owens Valley millimeter array has been used to map the CO 1-0 emission in the inner 25 of the grand-design spiral galaxy M51 at 2''-3'' resolution. These new images reveal the molecular spiral arms with unprecedented clarity; the emission in the two major arms (northeast and southwest) originates from supermassive cloud complexes, giant molecular associations (GMAs), which are for the first time resolved both along and perpendicular to the arms. The overall morphology of the CO emission is symmetric in reflection about the nucleus, with major complexes occurring opposite each other in the two major arms. On the other hand, the CO line flux in the area of the southwest arm closest to the nucleus is approximately twice as bright as that from the analogous location in the northeast arm. Streaming motions can be studied in detail and appear with great clarity along the major and minor axes of M51. The streaming velocities are very large, 60-150 km s^-1. Our maps offer, for the first time, sufficient resolution to resolve the structure in the molecular streaming motions. Both the radial and tangential velocity components show steep gradients, in qualitative accordance with predictions of the density-wave models of Roberts & Stewart. Our data thus support the presence of galactic shocks in the arms of M51. In general, velocity gradients across arms are higher by a factor of 2-10 than previously found. They vary in steepness along the spiral arms, becoming particularly steep in between GMAs. The steep gradients cause conditions of strong reverse shear in several regions in the arms, and thus the notion that shear is generally reduced by streaming motions in spiral arms will have to be modified. Of the three GMAs studied on the southwest arm, only one shows reduced shear. We find an unusual structure, an expansion in the northeast molecular arm at 25'' radius (1.2 kpc) southeast of the center. This broadening occurs right after the end of the northeast arm at the inner Lindblad resonance (ILR). Multiple-peak spectra, velocity twists, and structures with apparently high velocity dispersion are associated with this feature. Bifurcations in the molecular spiral arm structure, at a radius of ≈3.3 kpc (73''), may be evidence of a secondary compression of the gas caused by, and occurring near, the 4/1 ultraharmonic resonance. Several molecular spurs protrude from the main spiral arms, in particular on the western side of the map. Inside the radius of the ILR, we detect narrow (≈5'') molecular spiral arms, possibly related to the K-band arms found in the same region. We find evidence of noncircular motions in the inner 20'' of M51. The magnitude of these deviations is 20-30 km s^-1, and they are consistent with gas on elliptical orbits in a bar.