  The spin-orbit coupled lattice system under Zeeman fields provides an ideal
platform to realize exotic pairing states. Notable examples range from the
topological superfluid/superconducting (tSC) state, which is gapped in the bulk
but metallic at the edge, to the Fulde-Ferrell (FF) state (having a
phase-modulated order parameter with a uniform amplitude) and the
Larkin-Ovchinnikov (LO) state (having a spatially varying order parameter
amplitude). Here, we show that the topological FF state with Chern number
($\mathcal{C}=-1$) (tFF$_{1}$) and topological LO state with $\mathcal{C}=2$
(tLO$_{2}$) can be stabilized in Rashba spin-orbit coupled lattice systems in
the presence of both in-plane and out-of-plane Zeeman fields. Besides the
inhomogeneous tSC states, in the presence of a weak in-plane Zeeman field, two
topological BCS phases may emerge with $\mathcal{C}=-1$ (tBCS$_{1}$) far from
half filling and $\mathcal{C}=2$(tBCS$_{2}$) near half filling. We show
intriguing effects such as different spatial profiles of order parameters for
FF and LO states, the topological evolution among inhomogeneous tSC states, and
different non-trivial Chern numbers for the tFF$_{1}$ and tLO$_{1,2}$ states,
which are peculiar to the lattice system. Global phase diagrams for various
topological phases are presented for both half-filling and doped cases. The
edge states as well as local density of states spectra are calculated for tSC
states in a 2D strip.
