  The notion of a natural model of type theory is defined in terms of that of a
representable natural transfomation of presheaves. It is shown that such models
agree exactly with the concept of a category with families in the sense of
Dybjer, which can be regarded as an algebraic formulation of type theory. We
determine conditions for such models to satisfy the inference rules for
dependent sums, dependent products, and intensional identity types, as used in
homotopy type theory. It is then shown that a category admits such a model if
it has a class of maps that behave like the abstract fibrations in axiomatic
homotopy theory: they should be stable under pullback, closed under composition
and relative products, and there should be weakly orthogonal factorizations
into the class. It follows that many familiar settings for homotopy theory also
admit natural models of the basic system of homotopy type theory.
