The invention features a catheter and methods for delivering drug to tissue at a desired location of the wall of a body lumen. The catheter is constructed for insertion in a body lumen and has a catheter shaft and an expandable portion mounted on the catheter shaft. The expandable portion is expandable to a controlled pressure to fill the cross-section of the body lumen and press against the wall of the body lumen. In one embodiment, at least a portion of the exterior surface of the expandable portion is defined by a coating of a tenaciously adhered swellable hydrogel polymer. Incorporated in the hydrogel polymer is an aqueous solution of a preselected drug to be delivered to the tissue or plaque. The hydrogel polymer and drug are selected to allow rapid release of a desired dosage of the drug from the hydrogel polymer coating during compression of the hydrogel polymer coating against the wall of the lumen when the expandable portion is expanded. In other embodiments the polymer is released from the expandable portion in response to pressure, to coat the wall of the body lumen.