  We give a definition of finitary type theories that subsumes many examples of
dependent type theories, such as variants of Martin-L\"of type theory, simple
type theories, first-order and higher-order logics, and homotopy type theory.
We prove several general meta-theorems about finitary type theories: weakening,
admissibility of substitution and instantiation of metavariables, derivability
of presuppositions, uniqueness of typing, and inversion principles.
  We then give a second formulation of finitary type theories in which there
are no explicit contexts. Instead, free variables are explicitly annotated with
their types. We provide translations between finitary type theories with and
without contexts, thereby showing that they have the same expressive power. The
context-free type theory is implemented in the nucleus of the Andromeda 2 proof
assistant.
