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Direct numerical simulations of stably stratified, turbulent channel flow at low to moderate Reynolds number have been performed using large computational boxes and considering a wide range of stratification levels. For weak stratification or high Reynolds number, the turbulence is affected by buoyancy in the core of the channel, but the near-wall region differs little from the neutral case. With strong stratification, large laminar patches appear in the near-wall region and turbulent momentum and buoyancy fluxes vanish in the core of the channel. With increasing stratification, the near-wall streaks remain essentially unmodified, while large-scale global modes are damped. In the central region, internal gravity waves are dominant. In addition, there is an intermediate outer layer where the dynamics of the turbulent structures is governed by local fluxes. In this region, energy spectra collapse when using local Obukhov scaling.