This thesis describes the fabrication and use of nanostructures to study the physical properties of individual DNA molecules. We report DNA translocation experiments through solid-state nanopores with a diameter of about 10 nm. DNA ranging in length from 2000 to 48000 base pairs was detected and we find that the translocation time scales as a power law with length. Moreover, we show that such nanopores are an excellent tool for DNA size discrimination. To make suitable pores, we have developed a new fabrication technique. We show that a transmission electron microscope can be used to shrink pores in silicon oxide, with direct visual feedback. We conclude that the silicon oxide is softened by the electron beam, and deforms driven by its surface tension. Another subject of this thesis deals with the electrical transport properties of individual DNA molecules in the dry state. We have fabricated metal electrodes spaced by 50 to 500 nm on silicon oxide. DNA molecules were deposited between these electrodes, as observed with atomic force microscopy. We varied key parameters such as sequence, electrode material and substrate and conclude that DNA is insulating at the 50 nm length scale. Finally, this thesis explores the use of nanofabrication to obtain nanometer-sized electrodes for electrochemical experiments. We have fabricated electrodes by depositing metal on one side of the nanopore devices. A total of 15 devices with lateral dimensions ranging from 15 nm to 200 nm was characterized. We find that our data cannot be understood with a simple planar-disk electrode model. Much better agreement is found with simulations that take the precise electrode geometry into account. We conclude that the planar-disk model is very unreliable to estimate sizes of nanoelectrodes.