The application of graph analytics to various domains has yielded tremendous societal and economical benefits in recent years. However, the increasingly widespread adoption of graph analytics comes with a commensurate increase in the need to protect private information in graph data, especially in light of the many privacy breaches in real-world graph data that was supposed to preserve sensitive information. This paper provides a comprehensive survey of private graph data release algorithms that seek to achieve the fine balance between privacy and utility, with a specific focus on provably private mechanisms. Many of these mechanisms are natural extensions of the Differential Privacy framework to graph data, but we also investigate more general privacy formulations like Pufferfish Privacy that address some of the limitations of Differential Privacy. We also provide a wide-ranging survey of the applications of private graph data release mechanisms to social networks, finance, supply chain, and health care. This survey paper and the taxonomy it provides should benefit practitioners and researchers alike in the increasingly important area of private analytics and data release.