  We study the solubility of cubic equations over the integers. Assuming a
necessary congruence condition, the existence of such solutions is established
when the $h$-invariant of $C$ is at least $14$, improving on work of
Davenport-Lewis and generalizing the method from Heath-Brown's seminal work in
the homogeneous case. We also provide an upper bound on the smallest solution,
polynomially in the height of the coefficients. The method also yields new
results in the homogeneous case where we generalize and improve on previous
work of Browning, Dietmann and Elliott.
