  We develop algebraic models of simple type theories, laying out a framework
that extends universal algebra to incorporate both algebraic sorting and
variable binding. Examples of simple type theories include the unityped and
simply-typed $\lambda$-calculi, the computational $\lambda$-calculus, and
predicate logic.
  Simple type theories are given models in presheaf categories, with structure
specified by algebras of polynomial endofunctors that correspond to natural
deduction rules. Initial models, which we construct, abstractly describe the
syntax of simple type theories. Taking substitution structure into
consideration, we further provide sound and complete semantics in structured
cartesian multicategories. This development generalises Lambek's correspondence
between the simply-typed $\lambda$-calculus and cartesian-closed categories, to
arbitrary simple type theories.
