Recent studies of the performance of radio-frequency (rf) copper cavities operated at cryogenic temperatures have shown a dramatic increase in the maximum achievable surface electric field. We propose to exploit this development to enable a new generation of photoinjectors operated at cryogenic temperatures that may attain, through enhancement of the launch field at the photocathode, a significant increase in five-dimensional electron beam brightness. We present detailed studies of the beam dynamics associated with such a system, by examining an S-band photoinjector operated at $250\text{ }\text{ }\mathrm{MV}/\mathrm{m}$ peak electric field that reaches normalized emittances in the 40 nm-rad range at charges (100--200 pC) suitable for use in a hard x-ray free-electron laser (XFEL) scenario based on the LCLS. In this case, we show by start-to-end simulations that the properties of this source may give rise to high efficiency operation of an XFEL, and permit extension of the photon energy reach by an order of magnitude, to over 80 keV. The brightness needed for such XFELs is achieved through low source emittances in tandem with high current after compression. In the XFEL examples analyzed, the emittances during final compression are preserved using microbunching techniques. Extreme low emittance scenarios obtained at pC charge, appropriate for significantly extending temporal resolution limits of ultrafast electron diffraction and microscopy experiments, are also reviewed. While the increase in brightness in a cryogenic photoinjector is mainly due to the augmentation of the emission current density via field enhancement, further possible increases in performance arising from lowering the intrinsic cathode emittance in cryogenic operation are also analyzed. Issues in experimental implementation, including cavity optimization for lowering cryogenic thermal dissipation, external coupling, and cryocooler system, are discussed. We identify future directions in ultrahigh field cryogenic photoinjectors, including scaling to higher frequency, use of novel rf structures, and enabling of an extremely compact hard x-ray FEL.