Systems and methods for delivering drugs or other substances to target locations outside of blood vessel lumens or other body lumens. In some embodiments, a catheter having a penetrator is advanced into the body lumen and the penetrator is then advanced from the catheter toward a target location outside of that body lumen. The substance is then delivered through a the penetrator itself or through a delivery catheter that has been advanced through the penetrator to the target location. In other embodiments, the blood vessel or other body lumen is blocked at spaced-apart first and second locations and a quantity of the substance is introduced into the closed-off body lumen between the blocked first and second locations. The substance then diffuses into or through the wall of the vessel or body conduit to reach the target location.