Suppose that we have $r$ sensors and each one intends to send a function $z_i$
(e.g. a signal or an image) to a receiver common to all $r$ sensors. During transmission,
each $z_i$ gets convolved with a function $g_i$. The receiver records the function $y$,
given by the sum of all these convolved signals. When and under which conditions is it
possible to recover the individual signals $z_i$ and the blurring functions $g_i$ from just
one received signal $y$? This challenging problem, which intertwines blind deconvolution
with blind demixing, appears in a variety of applications, such as audio processing, image
processing, neuroscience, spectroscopy, and astronomy. It is also expected to play a
central role in connection with the future Internet-of-Things. We will prove that under
reasonable and practical assumptions, it is possible to solve this otherwise highly
ill-posed problem and recover the $r$ transmitted functions $z_i$ and the impulse responses
$g_i$ in a robust, reliable, and efficient manner from just one single received function
$y$ by solving a semidefinite program. We derive explicit bounds on the number of
measurements needed for successful recovery and prove that our method is robust in presence
of noise. Our theory is actually a bit pessimistic, since numerical experiments demonstrate
that, quite remarkably, recovery is still possible if the number of measurements is close
to the number of degrees of freedom.