Different contact materials and optimization of techniques of their depositions expand the possibilities to obtain high performance room temperature CdTe-based X/γ-ray detectors. The heterostructures with ohmic (MoOx) and Schottky (MoOx, TiOx, TiN, and In) contacts, created by DC reactive magnetron sputtering and vacuum thermal evaporation, as well as In/CdTe/Au diodes with a p-n junction, formed by laser-induced doping, have been developed and investigated. Depending on the surface pre-treatment of semi-insulating p-CdTe crystals, the deposition of a MoOx film formed either ohmic or Schottky contacts. Based on the calculations and I‑V characteristics of the Mo-MoOx/р-CdTe/MoOx-Mo, In/р-CdTe/MoOx-Mo, Ti-TiOx/р-CdTe/MoOx-Mo, and Ti-TiN/р-CdTe/MoOx-Mo Schottky-diode detectors, the current transport processes were described in the models of the carrier generation–recombination within the space-charge region (SCR) at low bias, and space-charge limited current incorporating the Poole–Frenkel effect at higher voltages, respectively. The energies of generation–recombination centers, density of trapping centers, and effective carrier lifetimes were determined. Nanosecond laser irradiation of the In electrode, pre-deposited on the p-CdTe crystals, resulted in extending the voltage range, corresponding to the carrier generation–recombination in the SCR in the I‑V characteristics of the In/CdTe/Au diodes. Such In/CdTe/Au p-n junction diode detectors demonstrated high energy resolutions (7%@59.5 keV, 4%@122 keV, and 1.6%@662 keV).