Cooling electrons in a nanoelectronic device to a few milikelvin, and further into the microkelvin regime, is a longstanding challenge. Weak electron-phonon coupling at low temperatures creates a bottleneck in traditional cooling techniques. Here we will present our approach to solving the problem: nuclear demagnetization refrigeration of on-chip copper to directly cool the electrons without intervening phonons. Our method has achieved a base electron temperature below 1.3 mK, held for several 1000s. On-chip refrigeration could potentially provide improvements in the operation of quantum simulators, computers and metrology standards, and open a new regime for the study of electron transport in nanostructures and 2D materials. However, it will be necessary to couple the copper refrigerant to the system of interest and to employ non-invasive thermometry techniques. We will discuss our progress towards these goals, including the development of a single-electron thermometer that is measured by RF reflectometry and does not require galvanic connection between the cooled electron gas and the outside world. This research is supported by the U.K. EPSRC (EP/N019199/1).