Crystal structure prediction (CSP) methods recently proposed a series of new rare-earth (RE) hydrides at high pressures with novel crystal structures, unusual stoichiometries, and intriguing features such as high-T-c superconductivity. RE trihydrides (REH3) generally undergo a phase transition from ambient P6(3)/mmc or P (3) over bar c1 to Fm (3) over barm at high pressure. This cubic REH3 (Fm (3) over barm) was considered to be a precursor to further synthesize RE polyhydrides such as YH4, YH6, YH9, and CeH9 with higher hydrogen contents at higher pressures. However, the structural stability and equation of state (EOS) of any of the REH3 have not been fully investigated at sufficiently high pressures. This work presents high-pressure X-ray diffraction (XRD) measurements in a laser-heated diamond anvil cell up to 100 GPa and ab initio evolutionary CSP of stable phases of DyH3 up to 220 GPa. Experiments observed the Fm (3) over barm phase of DyH3 to be stable at pressures from 17 to 100 GPa and temperatures up to similar to 2000 K. After complete decompression, the P (3) over bar c1 and Fm (3) over barm phases of DyH3 recovered under ambient conditions. Our calculations predicted a series of phases for DyH3 at high pressures with the structural phase transition sequence P (3) over bar c1 -> Imm2 Fm (3) over barm -> Pnma -> P6(3)/mmc at 11, 35, 135, and 194 GPa, respectively. The predicted P (3) over bar c1 and Fm (3) over barm phases are consistent with experimental observations. Furthermore, electronic band structure calculations were carried out for the predicted phases of DyH3, including the 4f states, within the DFT+U approach. The inclusion of 4f states shows significant changes in electronic properties, as more Dy d states cross the Fermi level and overlap with H 1s states. The structural phase transition from P (3) over bar c1 to Fm (3) over barm observed in DyH3 is systematically compared with other REH3 compounds at high pressures. The phase transition pressure in REH3 shows an inverse relation with the ionic radius of RE atoms.