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Most programs compiled to WebAssembly (Wasm) today are written in unsafe languages like C and C++. Unfortunately, memory-unsafe C code remains unsafe when compiled to Wasm -- and attackers can exploit buffer overflows and use-after-frees in Wasm almost as easily as they can on native platforms. Memory-Safe WebAssembly (MSWasm) proposes to extend Wasm with language-level memory-safety abstractions to precisely address this problem. In this paper, we build on the original MSWasm position paper to realize this vision. We give a precise and formal semantics of MSWasm, and prove that well-typed MSWasm programs are, by construction, robustly memory safe. To this end, we develop a novel, language-independent memory-safety property based on colored memory locations and pointers. This property also lets us reason about the security guarantees of a formal C-to-MSWasm compiler -- and prove that it always produces memory-safe programs (and preserves the semantics of safe programs). We use these formal results to then guide several implementations: Two compilers of MSWasm to native code, and a C-to-MSWasm compiler (that extends Clang). Our MSWasm compilers support different enforcement mechanisms, allowing developers to make security-performance trade-offs according to their needs. Our evaluation shows that the overhead of enforcing memory safety in software ranges from 22% (enforcing spatial safety alone) to 198% (enforcing full memory safety) on the PolyBenchC suite. More importantly, MSWasm's design makes it easy to swap between enforcement mechanisms; as fast (especially hardware-based) enforcement techniques become available, MSWasm will be able to take advantage of these advances almost for free.