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Meta-programming, writing programs that write other programs, involves two kinds of languages. The meta-language is the language in which meta-programs, which construct or manipulate other programs, are written. The object-language is the language of programs being manipulated. We study a class of meta-language features that are used to write meta-programs that are statically guaranteed to maintain semantic invariants of object-language programs, such as typing and scoping. We use type equality in the type system of the meta-language to check and enforce these invariants. Our main contribution is the illustration of the utility of type equality in typed functional meta-programming. In particular, we encode and capture judgments about many important language features using type equality. Finally, we show how type equality is incorporated as a feature of the type system of a practical functional meta-programming language. The core of this thesis is divided into three parts. First, we design a meta-programming language with dependent types. We use dependent types to ensure that well-typed meta-programs manipulate only well-typed object-language programs. Using this meta-language, we then construct highly efficient and safe interpreters for a strongly typed object language. We also prove the type safety of the meta-language. Second, we demonstrate how the full power of dependent types is not necessary to encode typing properties of object-languages. We explore a meta-language consisting of the programming language Haskell and a set of techniques for encoding type equality. In this setting we are able to carry out essentially the same meta-programming examples. We also expand the range of object-language features in our examples (e.g., pattern matching). Third, we design a meta-language (called Omega) with built-in equality proofs. This language is a significant improvement for meta-programming over Haskell: Omega's type system automatically manipulates proofs of type equalities in meta-programs. We further demonstrate our encoding and meta-programming techniques by providing representations and interpreters for object-languages with explicit substitutions and modal type systems.