  General quantum gravity arguments predict that Lorentz symmetry might not
hold exactly in nature. This has motivated much interest in Lorentz breaking
gravity theories recently. Among such models are vector-tensor theories with
preferred direction established at every point of spacetime by a fixed-norm
vector field. The dynamical vector field defined in this way is referred to as
the \textquotedblleft aether." In this paper, we put forward the idea of a
\textit{null} aether field and introduce, for the first time, the Null Aether
Theory (NAT)--a vector-tensor theory. We first study the Newtonian limit of
this theory and then construct exact spherically symmetric black hole solutions
in the theory in four dimensions, which contain Vaidya-type non-static
solutions and static Schwarzschild-(A)dS type solutions,
Reissner-Nordstr\"{o}m-(A)dS type solutions and solutions of conformal gravity
as special cases. Afterwards, we study the cosmological solutions in NAT: We
find some exact solutions with perfect fluid distribution for spatially flat
FLRW metric and null aether propagating along the $x$ direction. We observe
that there are solutions in which the universe has big-bang singularity and
null field diminishes asymptotically. We also study exact gravitational wave
solutions--AdS-plane waves and $pp$-waves--in this theory in any dimension
$D\geq3$. Assuming the Kerr-Schild-Kundt class of metrics for such solutions,
we show that the full field equations of the theory are reduced to two, in
general coupled, differential equations when the background metric assumes the
maximally symmetric form. The main conclusion of these computations is that the
spin-0 aether field acquires a \textquotedblleft mass" determined by the
cosmological constant of the background spacetime and the Lagrange multiplier
given in the theory.
