  Unimodular gravity is a compelling modified theory of gravity that offers a
natural solution to the cosmological constant problem. However, for unimodular
gravity to be considered a viable theory of gravity, one has to show that it
has a well-posed initial value formulation. Working in vacuum, we apply Dirac's
algorithm to find all the constraints of the theory. Then we prove that, for
initial data compatible with these constraints, the evolution is well posed.
Finally, we find sufficient conditions for a matter action to preserve the
well-posedness of the initial value problem of unimodular gravity. As a
corollary, we argue that the "unimodular" restriction on the spacetime volume
element can be satisfied by a suitable choice of the lapse function.
