The paper extends the usual world-volume action for a Dp-brane to the case of N coincident Dp-branes, where the world-volume theory involves a U(N) gauge theory. The construction is guided by the principle that the action should be consistent with T-duality. The resulting action includes various nonderivative interactions for the nonabelian scalar fields and shows that Dp-branes naturally couple to RR potentials of all form degrees, including both larger and smaller than p+1. The dynamics of Dp-branes moving in nontrivial background fields is discussed, and it is shown that these branes can be "polarized" by external fields. A simple example illustrates how a system of D0-branes in an external RR four-form field expands into a noncommutative two-sphere, forming a spherical D2-D0 bound state. The paper also provides a consistent nonabelian extension of the Born-Infeld and Chern-Simons actions, and compares the results with those derived from matrix theory.The paper extends the usual world-volume action for a Dp-brane to the case of N coincident Dp-branes, where the world-volume theory involves a U(N) gauge theory. The construction is guided by the principle that the action should be consistent with T-duality. The resulting action includes various nonderivative interactions for the nonabelian scalar fields and shows that Dp-branes naturally couple to RR potentials of all form degrees, including both larger and smaller than p+1. The dynamics of Dp-branes moving in nontrivial background fields is discussed, and it is shown that these branes can be "polarized" by external fields. A simple example illustrates how a system of D0-branes in an external RR four-form field expands into a noncommutative two-sphere, forming a spherical D2-D0 bound state. The paper also provides a consistent nonabelian extension of the Born-Infeld and Chern-Simons actions, and compares the results with those derived from matrix theory.