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We consider a generalization of the selective traveling salesman problem (STSP) in which the benefit of visiting a location changes over time. This new problem, called the selective travelling salesman problem with time-dependent profits (STSP-TDP), is defined on a graph with time-dependent profits associated with the vertices, and consists of determining a circuit of maximal total profit. In the STSP-TDP the tour length must not exceed a maximum value, and its starting and ending times must both lie within a prespecified planning horizon. This problem arises in planning tourist itineraries, mailbox collection, military surveillance, and water sampling, where the traveler accumulates different profits upon visiting the locations throughout the day. We focus on analyzing several variants of the problem depending on the shape of the time-dependent profit function. If this function is not monotonic, it may be worth visiting a …