The past decade has seen significant progress on the direct detection and characterization of young, self-luminous giant planets at wide orbital separations from their host stars. Some of these planets show evidence for disequilibrium processes like transport-induced quenching in their atmospheres; photochemistry may also be important, despite the large orbital distances. These disequilibrium chemical processes can alter the expected composition, spectral behavior, thermal structure, and cooling history of the planets, and can potentially confuse determinations of bulk elemental ratios, which provide important insights into planet-formation mechanisms. Using a thermo/photochemical kinetics and transport model, we investigate the extent to which disequilibrium chemistry affects the composition and spectra of directly imaged giant exoplanets. Results for specific "young Jupiters" such as HR 8799 b and 51 Eri b are presented, as are general trends as a function of planetary effective temperature, surface gravity, incident ultraviolet flux, and strength of deep atmospheric convection. We find that quenching is very important on young Jupiters, leading to CO/CH4 and N2/NH3 ratios much greater than, and H2O mixing ratios a factor of a few less than, chemical-equilibrium predictions. Photochemistry can also be important on such planets, with CO2 and HCN being key photochemical products. Carbon dioxide becomes a major constituent when stratospheric temperatures are low and recycling of water via the H2 + OH reaction becomes kinetically stifled. Young Jupiters with effective temperatures ≲700 K are in a particularly interesting photochemical regime that differs from both transiting hot Jupiters and our own solar-system giant planets.