  We present an analysis of MUSE observations obtained on the massive Frontier
Fields cluster Abell 2744. This new dataset covers the entire multiply-imaged
region around the cluster core. We measure spectroscopic redshifts for
HST-selected continuum sources together with line emitters blindly detected in
the datacube. The combined catalog consists of 514 spectroscopic redshifts
(with 414 new identifications), including 156 cluster members and 326 magnified
background sources. We use this redshift information to perform a
strong-lensing analysis of all multiple images previously found in the deep
Frontier Field images, and add three new MUSE-detected multiply-imaged systems
with no obvious HST counterpart. The combined strong lensing constraints
include a total of 60 systems producing 188 images altogether, out of which 29
systems and 83 images are spectroscopically confirmed, making Abell 2744 one of
the most well-constrained clusters to date. A parametric mass model including
two cluster-scale components in the core and several group-scale substructures
at larger radii accurately reproduces all the spectroscopic multiple systems,
reaching an rms of 0.67" in the image plane. Overall, the large number of
spectroscopic redshifts gives us a robust model and we estimate the systematics
on the mass density and magnification within the cluster core to be typically
~9%.
