In this paper, we develop distribution-free models that solve the appointment sequencing and scheduling problem by assuming only moments information of job durations. We show that our min-max appointment scheduling models, which minimize the worst-case expected waiting and overtime costs out of all probability distributions with the given marginal moments, can be exactly formulated as tractable conic programs. These formulations are obtained by exploiting hidden convexity of the problem. In the special case where only the first two marginal moments are given, the problem can be reformulated as a second-order cone program. Based on the structural properties of this formulation, under a mild condition, we derive the optimal time allowances in closed form and prove that it is optimal to sequence jobs in increasing order of job duration variance. We also prove similar results regarding the optimal time allowances and sequence for the case where only means and supports of job durations are known. This paper was accepted by Dimitris Bertsimas, optimization .