Stateless model checking is a useful state-space exploration technique for systematically testing complex real-world software. Existing stateless model checkers are limited to the verification of safety properties on terminating programs. However, realistic concurrent programs are nonterminating, a property that significantly reduces the efficacy of stateless model checking in testing them. Moreover, existing stateless model checkers are unable to verify that a nonterminating program satisfies the important liveness property of livelock-freedom, a property that requires the program to make continuous progress for any input. To address these short comings, this paper argues for incorporating a fair scheduler in stateless exploration. The key contribution of this paper is an explicit scheduler that is (strongly) fair and at the same time sufficiently nondeterministic to guarantee full coverage of safety properties. We have implemented the fair scheduler in the CHESS modelchecker. We show through theoretical arguments and empirical evaluation that our algorithm satisfies two important properties: 1) it visits all states of a finite-state program achieving state coverage at a faster rate than existing techniques, and 2) it finds all livelocks in a finite-state program. Before this work, nonterminating programs had to be manually modified in order to apply CHESS to them. The addition of fairness has allowed CHESS to be effectively applied to real-world nonterminating programs without any modification. For example, we have successfully booted the Singularity operating system under the control of CHESS.