Fuzzing is one of the key techniques for evaluating the robustness of programs against attacks. Fuzzing has to be effective in producing inputs that cover functionality and find vulnerabilities. But it also has to be efficient in producing such inputs quickly. Random fuzzers are very efficient, as they can quickly generate random inputs; but they are not very effective, as the large majority of inputs generated is syntactically invalid. Grammar-based fuzzers make use of a grammar (or another model for the input language) to produce syntactically correct inputs, and thus can quickly cover input space and associated functionality. Existing grammar-based fuzzers are surprisingly inefficient, though: Even the fastest grammar fuzzer Dharma still produces inputs about a thousand times slower than the fastest random fuzzer. So far, one can have an effective or an efficient fuzzer, but not both. In this paper, we describe how to build fast grammar fuzzers from the ground up, treating the problem of fuzzing from a programming language implementation perspective. Starting with a Python textbook approach, we adopt and adapt optimization techniques from functional programming and virtual machine implementation techniques together with other novel domain-specific optimizations in a step-by-step fashion. In our F1 prototype fuzzer, these improve production speed by a factor of 100--300 over the fastest grammar fuzzer Dharma. As F1 is even 5--8 times faster than a lexical random fuzzer, we can find bugs faster and test with much larger valid inputs than previously possible.