Homomorphic encryption, secure multi-party computation, and differential privacy are part of an emerging class of Privacy Enhancing Technologies which share a common promise: to preserve privacy whilst also obtaining the benefits of computational analysis. Due to their relative novelty, complexity, and opacity, these technologies provoke a variety of novel questions for design and governance. We interviewed researchers, developers, industry leaders, policymakers, and designers involved in their deployment to explore motivations, expectations, perceived opportunities and barriers to adoption. This provided insight into several pertinent challenges facing the adoption of these technologies, including: how they might make a nebulous concept like privacy computationally tractable; how to make them more usable by developers; and how they could be explained and made accountable to stakeholders and wider society. We conclude with implications for the development, deployment, and responsible governance of these privacy-preserving computation techniques.