Microneedle devices are provided for transport of therapeutic and biological molecules across tissue barriers. In a preferred embodiment, the microneedle devices permit drug delivery (or removal or sensing of body fluids) at clinically relevant rates across skin or other tissue barriers, without damage, pain, or irritation to the tissue. Methods of making these devices, which can include hollow and/or porous microneedles, are also provided. A preferred method for making a microneedle includes forming a micromold having sidewalls which define the outer surface of the microneedle, electroplating the sidewalls to form the hollow microneedle, and then removing the micromold from the microneedle. In a preferred method of use, the microneedle device is used to deliver fluid material into or across a biological barrier from one or more chambers in fluid connection with at least one of the microneedles. The device preferably further includes a means for controlling the flow of material through the microneedles. Representative examples of these means include the use of permeable membranes, fracturable impermeable membranes, valves, and pumps.