Over the past decade, a Radon-type transform called a conical Radon transform, which assigns to a given function its integral over various sets of cones, has arisen in the context of Compton cameras used in single photon emission computed tomography. Here, we study the conical Radon transform for which the central axis of the cones of integration is fixed. We present many of its properties, such as two inversion formulas, a stability estimate, and uniqueness and reconstruction for a local data problem. An existing inversion formula is generalized and a stability estimate is presented for general dimensions. The other properties are completely new results.