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.
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