  This is the first of a series of papers presenting the results from our
survey of 25 Galactic globular clusters with the MUSE integral-field
spectrograph. In combination with our dedicated algorithm for source
deblending, MUSE provides unique multiplex capabilities in crowded stellar
fields and allows us to acquire samples of up to 20 000 stars within the
half-light radius of each cluster. The present paper focuses on the analysis of
the internal dynamics of 22 out of the 25 clusters, using about 500 000 spectra
of 200 000 individual stars. Thanks to the large stellar samples per cluster,
we are able to perform a detailed analysis of the central rotation and
dispersion fields using both radial profiles and two-dimensional maps. The
velocity dispersion profiles we derive show a good general agreement with
existing radial velocity studies but typically reach closer to the cluster
centres. By comparison with proper motion data we derive or update the
dynamical distance estimates to 14 clusters. Compared to previous dynamical
distance estimates for 47 Tuc, our value is in much better agreement with other
methods. We further find significant (>3sigma) rotation in the majority (13/22)
of our clusters. Our analysis seems to confirm earlier findings of a link
between rotation and the ellipticities of globular clusters. In addition, we
find a correlation between the strengths of internal rotation and the
relaxation times of the clusters, suggesting that the central rotation fields
are relics of the cluster formation that are gradually dissipated via two-body
relaxation.
