Hydride molecules lie at the base of interstellar chemistry, but the synthesis of sulfuretted hydrides is poorly understood and their abundances often crudely constrained. Motivated by new observations of the Orion Bar photodissociation region (PDR) – 1″ resolution ALMA images of SH + ; IRAM 30 m detections of bright H 2 32 S, H 2 34 S, and H 2 33 S lines; H 3 S + (upper limits); and SOFIA/GREAT observations of SH (upper limits) – we perform a systematic study of the chemistry of sulfur-bearing hydrides. We self-consistently determine their column densities using coupled excitation, radiative transfer as well as chemical formation and destruction models. We revise some of the key gas-phase reactions that lead to their chemical synthesis. This includes ab initio quantum calculations of the vibrational-state-dependent reactions SH + + H 2 ( v ) ⇄ H 2 S + + H and S + H 2 ( v ) ⇄ SH + H. We find that reactions of UV-pumped H 2 ( v ≥ 2) molecules with S + ions explain the presence of SH + in a high thermal-pressure gas component, P th ∕ k ≈ 10 8 cm −3 K, close to the H 2 dissociation front (at A V < 2 mag). These PDR layers are characterized by no or very little depletion of elemental sulfur from the gas. However, subsequent hydrogen abstraction reactions of SH + , H 2 S + , and S atoms with vibrationally excited H 2 , fail to form enough H 2 S + , H 3 S + , and SH to ultimately explain the observed H 2 S column density (~2.5 × 10 14 cm −2 , with an ortho-to-para ratio of 2.9 ± 0.3; consistent with the high-temperature statistical value). To overcome these bottlenecks, we build PDR models that include a simple network of grain surface reactions leading to the formation of solid H 2 S (s-H 2 S). The higher adsorption binding energies of S and SH suggested by recent studies imply that S atoms adsorb on grains (and form s-H 2 S) at warmer dust temperatures ( T d < 50 K) and closer to the UV-illuminated edges of molecular clouds. We show that everywhere s-H 2 S mantles form(ed), gas-phase H 2 S emission lines will be detectable. Photodesorption and, to a lesser extent, chemical desorption, produce roughly the same H 2 S column density (a few 10 14 cm −2 ) and abundance peak (a few 10 −8 ) nearly independently of n H and G 0 . This agrees with the observed H 2 S column density in the Orion Bar as well as at the edges of dark clouds without invoking substantial depletion of elemental sulfur abundances.