  We develop the usage of certain type theories as specification languages for
algebraic theories and inductive types. We observe that the expressive power of
dependent type theories proves useful in the specification of more complicated
algebraic theories. We describe syntax and semantics for three classes of
algebraic theories: finitary quotient inductive-inductive theories, their
infinitary generalization, and finally higher inductive-inductive theories. In
each case, an algebraic signature is a typing context or a closed type in a
specific type theory.
