Bent crystals can be used to efficiently steer charged particles by trapping impacting particles in the potential well generated by adjacent crystalline planes, resulting in a deflection equivalent to the effect of a magnetic field of hundreds of Tesla. This impressive property stimulates accelerator physicists to make use of crystals for advanced beam manipulations. In this Ph.D. work, the main results gathered with the use of bent crystals at the high-energy frontier in the CERN accelerator complex are reported, with particular focus on crystal collimation tests carried out at the LHC in 2018 with Pb ion beams. These activities allowed to demonstrate the cleaning improvement provided by this collimation scheme, and were a key ingredient in the decision to include crystal collimation in the HL-LHC baseline for the run starting in 2022. The code development effort aimed at improving the available simulation tools for crystal collimation with proton beams is also described, using as case studies two important applications beyond ion beam collimation that were explored at CERN: the double-crystal setup at the SPS and the deployment of crystals for the reduction of background on forward physics detectors.