—Good programming languages provide helpful abstractions for writing secure code, but the security properties of the source language are generally not preserved when compiling a program and linking it with adversarial code in a low-level target language (e.g., a library or a legacy application). Linked target code that is compromised or malicious may, for instance, read and write the compiled program’s data and code, jump to arbitrary memory locations, or smash the stack, blatantly violating any source-level abstraction. By contrast, a fully abstract compilation chain protects source-level abstractions all the way down, ensuring that linked adversarial target code cannot observe more about the compiled program than what some linked source code could about the source program. However, while research in this area has so far focused on preserving observational equivalence, as needed for achieving full abstraction, there is a much larger space of security properties one can choose to preserve against linked adversarial code. And the precise class of security properties one chooses crucially impacts not only the supported security goals and the strength of the attacker model, but also the kind of protections a secure compilation chain has to introduce. We are the first to thoroughly explore a large space of formal secure compilation criteria based on robust property preservation, i.e., the preservation of properties satisfied against arbitrary adversarial contexts. We study robustly preserving various classes of trace properties such as safety, of hyperproperties such as noninterference, and of relational hyperproperties such as trace equivalence. This leads to many new secure compilation criteria, some of which are easier to practically achieve and prove than full abstraction, and some of which provide strictly stronger security guarantees. For each of the studied criteria we propose an equivalent “property-free” characterization that clarifies which proof techniques apply. For relational properties and hyperproperties, which relate the behaviors of multiple programs, our formal definitions of the property classes themselves are novel. We order our criteria by their relative strength and show several collapses and separation results. Finally, we adapt existing proof techniques to show that even the strongest of our secure compilation criteria, the robust preservation of all relational hyperproperties, is achievable for a simple translation from a statically typed to a dynamically typed language.