We introduce mixed twistor D-modules and establish their fundamental functorial properties. We
also prove that they can be described as the gluing of admissible variations of mixed twistor
structures. In a sense mixed twistor D-modules can be regarded as a twistor version of M.
Saito's mixed Hodge modules. Alternatively they can be viewed as a mixed version of the pure
twistor D-modules studied by C. Sabbah and the author. The theory of mixed twistor D-modules is
one of the ultimate goals in the study suggested by Simpson's Meta Theorem and it would form a
foundation for the Hodge theory of holonomic D-modules which are not necessarily regular
singular.
