In modern computer systems, jobs are divided into short tasks and executed in parallel. Empirical observations in practical systems suggest that the task service times are highly random and the job service time is bottlenecked by the slowest straggling task. One common solution for straggler mitigation is to replicate a task on multiple servers and wait for one replica of the task to finish early. The delay performance of replications depends heavily on the scheduling decisions of when to replicate, which servers to replicate on, and which job to serve first. So far, little is understood on how to optimize these scheduling decisions for minimizing the delay to complete the jobs. In this paper, we present a comprehensive study on delay-optimal scheduling of replications in both centralized and distributed multi-server systems. Low-complexity scheduling policies are designed and are proven to be delay-optimal or near delay-optimal in stochastic ordering among all causal and non-preemptive policies. These theoretical results are established for general system settings and delay metrics that allow for arbitrary arrival processes, arbitrary job sizes, arbitrary due times, and heterogeneous servers with data locality constraints. Novel sample-path tools are developed to prove these results.