  We introduce mixed twistor $D$-modules, and establish the fundamental
functorial property. We also prove that they are described as the gluing of
admissible variations of mixed twistor structure. In a sense, mixed twistor
$D$-modules could be regarded as a twistor version of mixed Hodge modules due
to M. Saito . In the other sense, they could be a mixed version of pure twistor
$D$-modules studied by C. Sabbah and the author. A theory of mixed twistor
$D$-modules is one of the ultimate goals in the study suggested by Simpson's
Meta Theorem, and the author hopes that it would play a basic role in the Hodge
theory for holonomic $D$-modules possibly with irregular singularity.
