Liquid xenon offers several features, which make it suitable for applications in nuclear medicine, such as high scintillation yield and fast scintillation decay time. Moreover, being a continuous medium with a uniform response, liquid xenon allows one to avoid most of the geometrical distortions of conventional detectors based on scintillating crystals. In this paper, we describe how these properties have motivated the development of a novel concept for positron emission tomography scanners with Time-Of-Flight measurement, which combines a liquid xenon scintillating volume and silicon photomultipliers as sensors. A Monte Carlo investigation has pointed out that this technology would provide an excellent intrinsic time resolution, down to 70 ps. Also, the transparency of liquid xenon to UV and blue wavelengths opens the possibility of exploiting both scintillation and Cherenkov light for a high-sensitivity positron emission tomography scanner with Time-Of-Flight capabilities. Monte Carlo simulations point to a time resolution of 30-50 ps obtained using Cherenkov light. A prototype is being built to demonstrate the high resolution in energy, time and reconstruction of spatial coordinates of this concept, using a ring of 30 cm internal diameter and a depth of 3 cm instrumented with VUV-sensitive silicon photomultipliers.