3C 396 is a composite supernova remnant (SNR), consisting of a central pulsar wind nebula (PWN) and a bright shell in the west, which is known to be interacting with molecular clouds (MCs). We present a study of X-ray emission from the shell and the PWN of the SNR 3C 396 using archival Suzaku data. The spectrum of the SNR shell is clearly thermal, without a signature of a non-thermal component. The abundances of Al and Ca from the shell are slightly enhanced, which indicates the presence of metal-enriched supernova ejecta. The PWN spectra are well described by a power-law model with a photon index of similar to 1.97 and a thermal component with an electron temperature of similar to 0.93 keV. The analysis of about 11 yr of Fermi data revealed an 18 sigma detection of gamma-ray emission from the location overlapping with the position of 3C 396 / 4FGL J1903.8+0531. The spectrum of 3C 396 / 4FGL J1903.8+0531 is best fitted with a log-parabola function with parameters of alpha = 2.66 and beta = 0.16 in the energy range of 0.2-300 GeV. The luminosity of 3C 396 / 4FGL J1903.8+0531 was found to be >10(35) erg s(-1) at 6.2 kpc, which rules out the inverse Compton emission model. Possible scenarios of gamma-ray emission are hadronic emission and bremsstrahlung processes, due to the fact that the SNR is expanding into dense MCs in the western and northern regions of the SNR.