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Research in concurrent in-memory data structures has focused almost exclusively on models where processes are either reliable, or may fail by crashing permanently. The case where processes may recover from failures has received little attention because recovery from conventional volatile memory is impossible in the event of a system crash, during which both the state of main memory and the private states of processes are lost. Future hardware architectures are likely to include various forms of non-volatile random access memory (NVRAM), creating new opportunities to design robust main memory data structures that can recover from system crashes. In this paper we advance the theoretical foundations of such data structures in two ways. First, we review several known variations of Herlihy and Wing's linearizability property that were proposed in the context of message passing systems but also apply in our NVRAM-based model, we discuss the limitations of these properties with respect to our specific goals, and we propose an alternative correctness condition called recoverable linearizability. Second, we discuss techniques for implementing shared objects that satisfy such properties with a focus on wait-free implementations. Specifically, we demonstrate how to achieve different variations of linearizability in our model by transforming two classic wait-free constructions.