The kinetic Alfv́n wave has been recognized as an important wave mode in magnetospheric plasmas and laboratory plasmas, and has potential application in many areas of cosmic plasma physics. The kinetic dispersion relation of this mode has been described including finite frequency and finite ion gyroradius corrections. Laboratory plasmas as well as plasmas in space often contain strong gradients perpendicular to the background magnetic field. In this case, the dispersion relation must be generalized to include changes in the plasma parameters on each side of the gradient. In the presence of such gradients, localized modes can be found in the plasma. Depending on the relative values of the Alfv́n speed and the plasma beta across these gradients, these modes can be trapped within the cavity or enhancement or propagate across the gradient.