We study the structure of the backward shift-invariant and nearly invariant subspaces in weighted Fock-type spaces whose weight is not necessarily radial. We show that in the spaces which contain polynomials as a dense subset (in particular, in the radial case) any nontrivial backward shift-invariant subspace coincides with a finite dimensional subspace consisting of polynomials up to a certain degree. In general, the structure of nearly invariant subspaces is more complicated. In the case of spaces of slow growth (up to zero exponential type) we establish an analogue of de Branges' Ordering Theorem. This is a joint work with Alexandru Aleman, Yurii Belov, and Haakan Hedenmalm.