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This master thesis presents a new type of Positron Emission TOF Apparatus using Liquid xenOn (PETALO). The detector is based in the Liquid Xenon Scintillating Cell (LXSC). The cell is a box filled with liquid xenon (LXe) whose transverse dimensions are chosen to optimize packing and with a thickness optimized to contain a large fraction of the incoming photons. The entry and exit faces of the box (relative to the incoming gammas direction) are instrumented with large silicon photomultipliers (SiPMs), coated with a wavelength shifter, tetraphenyl butadiene (TPB). The non-instrumented faces are covered by reflecting Teflon coated with TPB. In this thesis we show that the LXSC can display an energy resolution of 5% FWHM, much better than that of conventional solid scintillators such as LSO/LYSO. The LXSC can measure the interaction point of the incoming photon with a resolution in the three coordinates of 1 mm. The very fast scintillation time of LXe (2 ns) and the availability of suitable sensors and electronics permits a coincidence resolution time (CRT) in the range of 100-200 ps, again much better than any current PET-TOF system. The LXSC constitutes the core of a high-sensitivity, nuclear magnetic resonance compatible, PET device, with enhanced Time Of Flight (TOF) sensitivity.