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This article treats both discrete time and continuous time stopping problemsfor general Markov processes on the real line with general linear costs. Usingan auxiliary function of maximum representation type, conditions are given toguarantee the optimal stopping time to be of threshold type. The optimalthreshold is then characterized as the root of that function. For random walksour results condense in the fact that all combinations of concave increasingpay-off functions and convex cost functions lead to a one-sided solution. ForL\'evy processes an explicit way to obtain the auxiliary function and thethreshold is given by use of the ladder height processes. Lastly, theconnection from discrete and continuous problem and possible approximation ofthe latter one via the former one is discussed.

General Optimal Stopping with Linear Costs