A homogeneous fluid is bounded above and below by horizontal plane surfaces in rapid rotation about a vertical axis. An obstacle is attached to one of the surfaces, and at large distances from the obstacle the relative velocity is steady and horizontal. Solutions are obtained as power series expansions in the Rossby number, uniformly valid as the Taylor number approaches infinity. If the height of the obstacle is greater than the Rossby number times the depth, a stagnant region (Taylor column) forms over the obstacle. Outside this region there is a net circulation in a direction opposite the rotation. The shape of the stagnant region and the circulation are uniquely determined as part of the solution. Possible geophysical applications are discussed, and it is shown that stratification renders Taylor columns unlikely on earth, but that the Great Red Spot of Jupiter may be an example of this phenomenon, as Hide has suggested. Abstract A homogeneous fluid is bounded above and below by horizontal plane surfaces in rapid rotation about a vertical axis. An obstacle is attached to one of the surfaces, and at large distances from the obstacle the relative velocity is steady and horizontal. Solutions are obtained as power series expansions in the Rossby number, uniformly valid as the Taylor number approaches infinity. If the height of the obstacle is greater than the Rossby number times the depth, a stagnant region (Taylor column) forms over the obstacle. Outside this region there is a net circulation in a direction opposite the rotation. The shape of the stagnant region and the circulation are uniquely determined as part of the solution. Possible geophysical applications are discussed, and it is shown that stratification renders Taylor columns unlikely on earth, but that the Great Red Spot of Jupiter may be an example of this phenomenon, as Hide has suggested.