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Exceptions and errors occurring within mission critical applications due to hardware failures have a high cost. With the emerging Next Generation Platforms (NGPs), the rate of hardware failures will invariably increase. Therefore, designing our applications to be resilient is a critical concern in order to retain the reliability of results while meeting the constraints on power budgets. In this paper, we implement software resilience in HPX, an Asynchronous Many-Task Runtime system. We implement two resiliency APIs that we expose to the application developers, namely task replication and task replay. Task replication repeats a task n-times and executes them asynchronously. Task replay will reschedule a task up to n-times until a valid output is returned. Furthermore, we introduce an API that allows the application to verify the returned result with a user provided predicate. We test the APIs with both artificial workloads and a dataflow based stencil application. We demonstrate that only minor overheads are incurred when utilizing these resiliency features for work loads where the task size is greater than 200 $μ$s. We also show that most of the added execution time arises from the replay or replication of the tasks themselves and not by the implementation of the APIs.