The cesium polyhydrides (CsHn, n > 1) are predicted to become stable, with respect to decomposition into CsH and H-2, at pressures as low as 2 GPa. The CsH3 stoichiometry is found to have the lowest enthalpy of formation from CsH and H-2 between 30 and 200 GPa. Evolutionary algorithms predict five distinct, mechanically stable, nearly isoenthalpic CsH3 phases consisting of H-3(-) molecules and Cs+ atoms. The H-3(-) sublattices in two of these adopt a hexagonal three-connected net; in the other three the net is twisted, like the silicon sublattice in the alpha-ThSi2 structure. The former emerge as being metallic below 100 GPa in our screened hybrid density functional theory calculations, whereas the latter remain insulating up to pressures greater than 250 GPa. The Cs+ cations in the most stable I4(1)/amd CsH3 phase adopt the positions of the Cs atoms in Cs-IV, and the H-3(-) molecules are found in the (interstitial) regions which display a maximum in the electron density.