The theory of shift invariant subspaces on locally compact abelian groups has been extremely grown in last two decades. Let G be a locally compact abelian group with a closed, discrete and cocompact subgroup Γ. A closed subspace V of L2(G) is said to be shift invariant if it is invariant under shifts by elements of Γ. The omission of some conditions (discrete and cocompact) on Γ generalizes shift invariant spaces to translation invariant spaces. In this paper we briefly review this generalization. We also prove similar results on modulation invariant spaces which lead to characterization of Gabor- type invariant spaces.

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