The geometry of spinning codimension-two branes in anti-de Sitter (AdS) spacetime is analyzed in three and higher dimensions. The construction of nonextremal solutions is based on identifications in the covering of AdS space by isometries that have fixed points. The discussion focuses on the cases where the parameters of spinning states can be related to the velocity of a boosted static codimension-two brane. The resulting configuration describes a single spinning brane, or a set of intersecting branes, each one produced by an independent identification. The nature of the singularity is also examined, establishing that the AdS curvature acquires one in the form of a Dirac delta distribution. The stability of the branes is studied in the framework of Chern-Simons AdS supergravity. A class of branes, characterized by one free parameter, is shown to be stable when the Bogomol'nyi-Prasad-Sommerfeld (BPS) conditions are satisfied. In three dimensions, these stable branes are extremal, while in higher dimensions, the BPS branes are not the extremal ones.