  In this paper we develop a bilinearisation-reduction approach to derive
solutions to the classical and nonlocal nonlinear Schr\"{o}dinger (NLS)
equations with nonzero backgrounds. We start from the second order
Ablowitz-Kaup-Newell-Segur coupled equations as an unreduced system. With a
pair of solutions $(q_0,r_0)$ we bilinearize the unreduced system and obtain
solutions in terms of quasi double Wronskians. Then we implement reductions by
introducing constraints on the column vectors of the Wronskians and finally
obtain solutions to the reduced equations, including the classical NLS equation
and the nonlocal NLS equations with reverse-space, reverse-time and
reverse-space-time, respectively. With a set of plane wave solution $(q_0,r_0)$
as a background solution, we present explicit formulae for these column
vectors. As examples, we analyze and illustrate solutions to the focusing NLS
equation and the reverse-space nonlocal NLS equation. In particular, we present
formulae for the rouge waves of arbitrary order for the focusing NLS equation.
