In the Priority k-Center problem, the input consists of a metric space (X, d), an integer k and for each point v ∈ X a priority radius r(v). The goal is to choose k-centers S ⊆ X to minimize maxv∈X 1/r(v) d(v, S). If all r(v)'s were uniform, one obtains the classical k-center problem. Plesník  introduced this problem and gave a 2-approximation algorithm matching the best possible algorithm for vanilla k-center. We show how the Priority k-Center problem is related to two different notions of fair clustering [23, 28]. Motivated by these developments we revisit the problem and, in our main technical contribution, develop a framework that yields constant factor approximation algorithms for Priority k-Center with outliers. Our framework extends to generalizations of Priority k-Center to matroid and knapsack constraints, and as a corollary, also yields algorithms with fairness guarantees in the lottery model of Harris et al.