  We construct several new integrable systems corresponding to nonlocal
versions of the Hirota equation, which is a particular example of higher order
nonlinear Schr\"{o}dinger equations. The integrability of the new models is
established by providing their explicit forms of Lax pairs or zero curvature
conditions. The two compatibility equations arising in this construction are
found to be related to each other either by a parity transformation
$\mathcal{P}$, by a time reversal $\mathcal{T}$ or a
$\mathcal{PT}$-transformation possibly combined with a conjugation. We
construct explicit multi-soliton solutions for these models by employing
Hirota's direct method as well as Darboux-Crum transformations. The nonlocal
nature of these models allows for a modification of these solution procedures
as the new systems also possess new types of solutions with different parameter
dependence and different qualitative behaviour. The multi-soliton solutions are
of varied type, being for instance nonlocal in space, nonlocal in time of time
crystal type, regular with local structures either in time/space or of rogues
wave type.
