High time resolution is becoming increasingly important for many applications in nuclear medicine and high energy physics applications. The introduction of time-of-flight (TOF) into positron emission tomography (PET) has helped to improve noise properties of reconstructed PET images. TOF-PET systems with timing resolution in the order of 500-600 ps FWHM using detectors based on photomultiplier tubes (PMT) are commercially available from major PET vendors. Coincidence time resolutions (CTRs) at the level of 100 ps FWHM are needed to increase the signal-to-noise ratio of the reconstructed images in an extent that patients can benefit from shorter acquisition times and lower radiation exposure. Laboratory measurements already achieve this value by scintillation methods. This thesis aims at an increase of the time resolution of TOF-PET by investigation of the Cherenkov effect as an almost instantaneous process of luminescence. The Cherenkov effect occurs after the photoelectric absorption of the 511 keV annihilation photons inside the scintillator or Cherenkov radiator and provides very precise time information about the energy deposition. Furthermore, the development of silicon photomultipliers (SiPM), with properties such as single photon detection, good time resolution and-in contrast to ordinary PMTs-insensitivity to magnetic fields, allows their utilisation in hybrid devices such as PET combined with nuclear magnetic resonance (NMR) imaging. Recently, the digital SiPM was introduced, which exploits the quasi digital nature of SiPM and therefore provides advantages such as integrated readout of the data. To make use of these advantages, SiPM were used in this work for the investigation on the Cherenkov effect. First, the time resolution of SiPM, both analogue and digital are determined and compared using pulsed lasers in the pico- and femtosecond region. A Monte-Carlo simulation tool was developed for better understanding of the obtained results. Then, factors influencing the time resolution of scintillators and Cherenkov radiators are determined and compared. Using simulations it is shown that Cherenkov emission by electrons at energies below 500 keV can be expected in Cherenkov radiators and scintillators, although, the number of emitted Cherenkov photons is low. A study on the influence several parameters shows that the UV-transmission is the most important factor for increasing the number of detected Cherenkov photons. Although, the utilisation of pure Cherenkov radiators allows more flexibility on the material parameters, the low yield of Cherenkov photons makes a determination of the deposited energy almost impossible. Energy determination is necessary in PET for discriminating true events from coincidences after Compton scattering. As a consequence, the application of hybrid scintillators could provide both, very precise time resolution due to the Cherenkov emission and additional energy information due to scintillation. Therefore, a case study for LSO:Ce on the impact of additionally detecting Cherenkov photons was done and showed significant improvement of the CTR. Finally, proof of principle measurements are presented, showing the feasibility of detecting Cherenkov photons after the photoelectric absorption of 511 keV annihilation photons. It is shown that using the Philips digital photon counter (DPC) on average 5.7 Cherenkov photons can be detected for the inorganic crystal LuAG and 3.1 Cherenkov photons can be detected using the optical glass (N-LASF31A). Measuring a 8 mm long LuAG crystal as Cherenkov radiator in coincidence with LSO:Ce, a coincidence time resolution of 145 ± 6.2 ps FWHM could be achieved, which is significantly better than the result of the reference measurement of 192 ± 4.0 ps FWHM achieved for two LSO:Ce crystals with the same dimensions. Furthermore, in a coincidence measurement using two BGO scintillators 24% of the coincidences were found to be triggered by Cherenkov-photons. Utilising the Cherenkov emission in BGO, a coincidence time resolution of 301 ps FWHM could be achieved for two crystals with 8 mm length. The obtained CTR of the scintillation emission was 2.38 ns FWHM. For this setup an energy resolution of 16.7% was achieved, which proofs the feasibility of utilising fast Cherenkov photons for improvement of time resolution in scintillators while preserving the energy information.