We report a high-pressure study of monoclinic monazite-type SrCrO4 up to 26 GPa. Therein we combined x-ray diffraction, Raman and optical-absorption measurements with ab initio calculations, to find a pressure-induced structural phase transition of SrCrO4 near 8-9 GPa. Evidence of a second phase transition was observed at 10-13 GPa. The crystal structures of the high-pressure phases were assigned to the tetragonal scheelite-type and monoclinic AgMnO4-type structures. Both transitions produce drastic changes in the electronic band gap and phonon spectrum of SrCrO4. We determined the pressure evolution of the band gap for the low-pressure and high-pressure phases as well as the frequencies and pressure dependences of the Raman-active modes. In all three phases most Raman modes harden under compression; however the presence of low-frequency modes which gradually soften is also detected. In monazite-type SrCrO4, the band gap blue-shifts under compression, but the transition to the scheelite phase causes an abrupt decrease of the band gap in SrCrO4. Calculations showed good agreement with experiments and were used to better understand the experimental results. From x-ray diffraction studies and calculations we determined the pressure dependence of the unit-cell parameters of the different phases and their ambient-temperature equations of state. The results are compared with the high-pressure behavior of other monazites, in particular PbCrO4. A comparison of the high-pressure behavior of the electronic properties of SrCrO4 (SrWO4) and PbCrO4 (PbWO4) will also be made. Finally, the possible occurrence of a third structural phase transition is discussed. ; Comment: 41 pages, 9 figures, 7 tables