Methods and apparatus for forming apertures in a solid state membrane using dielectric breakdown are provided. In one disclosed arrangement a plurality of apertures are formed. The membrane comprises a first surface area portion on one side of the membrane and a second surface area portion on the other side of the membrane. Each of a plurality of target regions comprises a recess or a fluidic passage opening out into the first or second surface area portion. The method comprises contacting all of the first surface area portion of the membrane with a first bath comprising ionic solution and all of the second surface area portion with a second bath comprising ionic solution. A voltage is applied across the membrane via first and second electrodes in respective contact with the first and second baths comprising ionic solutions to form an aperture at each of a plurality of the target regions in the membrane.