Outpatient appointment system design is a complex problem because it involves multiple stakeholders, sequential booking process, random arrivals, no-shows, varying degrees of urgency of different patients’ needs, service time variability, and patient and provider preferences. Clinics use a two-step process to manage appointments. In the first step, which we refer to as the clinic profile setup problem, service providers’ daily clinic time is divided into appointment slots. In the second step, which we refer to as the appointment booking problem, physicians’ offices decide which available slots to book for each incoming request for an appointment. In this chapter, we present formulations of mathematical models of key problems in the area of appointment system design. We also discuss the challenges and complexities of solving such problems. In addition, summaries of prior research, particularly advanced models related to the examples shown in this chapter are also presented.