  We introduce type-theoretic algebraic weak factorisation systems and show how
they give rise to homotopy-theoretic models of Martin-L\"of type theory. This
is done by showing that the comprehension category associated to a
type-theoretic algebraic weak factorisation system satisfies the assumptions
necessary to apply a right adjoint method for splitting comprehension
categories. We then provide methods for constructing several examples of
type-theoretic algebraic weak factorisation systems, encompassing the existing
groupoid model and cubical sets models, as well as some models based on normal
fibrations
