This dissertation investigates the behavior of cold dark matter (CDM) on galaxy scales. We present well-resolved Halpha velocity fields of the central regions of 17 dark matter-dominated low surface brightness (LSB) and dwarf galaxies observed with the DensePak Integrated Field Unit. We derive rotation curves from the two-dimensional data and compare them to published long-slit and HI rotation curves. We find broad consistency between the independent data sets. Under several assumptions about the velocity contribution from the baryons, we fit the dark matter component with cuspy NFW and cored pseudoisothermal halos. We find the data to be better described by cored dark matter halos. For the majority of galaxies, NFW halo fits either cannot be made or the implied concentrations are too low for LCDM. The shapes of the NFW rotation curves are also inconsistent with the galaxy rotation curves. We find that CDM predicts a substantial cusp mass excess near the centers of the galaxies and that the ratio of predicted to observed dark matter increases as baryons become more important. We investigate claims that systematic effects including beam smearing, slit misplacement and noncircular motions are responsible for slowly rising long-slit and HI rotation curves. We find the DensePak rotation curves to also be slowly rising, supporting the idea that this is an intrinsic feature of LSB rotation curves. We also model the two-dimensional NFW halo and test several modifications to the potential in an attempt to simultaneously reconcile both the NFW velocity field and rotation curve with observed galaxy data. We present mock DensePak velocity fields and rotation curves of axisymmetric and non-axisymmetric potentials. We find that a non-axisymmetric NFW potential with a constant axis ratio can reduce the cusp mass excess in the observed galaxy data, but the observer's line-of-sight must be along the minor axis of the potential, and the NFW pinch is not erased from the velocity field. We find that a non-axisymmetric NFW potential with a radially varying axis ratio tends to wash out the NFW pinch but introduces a twist to the velocity field.