There is growing recognition that machine learning (ML) exposes new security and privacy vulnerabilities in software systems, yet the technical community's understanding of the nature and extent of these vulnerabilities remains limited but expanding. In this talk, we explore the threat model space of ML algorithms through the lens of Saltzer and Schroeder's principles for the design of secure computer systems. This characterization of the threat space prompts an investigation of current and future research directions. We structure our discussion around three of these directions, which we believe are likely to lead to significant progress. The first encompasses a spectrum of approaches to verification and admission control, which is a prerequisite to enable fail-safe defaults in machine learning systems. The second seeks to design mechanisms for assembling reliable records of compromise that would help understand the degree to which vulnerabilities are exploited by adversaries, as well as favor psychological acceptability of machine learning applications. The third pursues formal frameworks for security and privacy in machine learning, which we argue should strive to align machine learning goals such as generalization with security and privacy desiderata like robustness or privacy. Key insights resulting from these three directions pursued both in the ML and security communities are identified and the effectiveness of approaches are related to structural elements of ML algorithms and the data used to train them. We conclude by systematizing best practices in our community.