We study conditions for the existence of stable, strategy-proof mechanisms in a many-to-one matching model with salaries. Workers and rms want to match and agree on the terms of their match. Firms demand different sets of workers at different salaries. Workers have preferences over different firm-salary combinations. Workers' preferences are monotone in salaries. We show that for this model, a descending auction mechanism is the only candidate for a stable mechanism that is strategy-proof for workers. Moreover, we identify a maximal domain of demand functions for firms, such that the mechanism is stable and strategy-proof. In the special case, where demand functions are generated by quasi-linear profit functions, our domain reduces to the domain of demand functions under which workers are gross substitutes for firms. We provide two versions of the results for the quasi-linear case. One for a discrete model, where salaries are restricted to discrete units and one for a continuous model, where salaries can take on arbitrary positive numbers. More generally, we show that several celebrated results (the existence of a worker-optimal stable allocation, the rural hospitals theorem, the strategyproofness of the worker-optimal stable mechanism) generalize from the discrete to the continuous model.