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We say that a sequence of proper geodesic spaces $X_i$ consists of\textit{almost homogeneous spaces} if there is a sequence of discrete groups ofisometries $G_i \leq Iso(X_i)$ such that diam$(X_i/G_i)\to 0$ as $i \to\infty$.  We show that if a sequence $X_i$ of almost homogeneous spaces converges inthe pointed Gromov--Hausdorff sense to a space $X$, then $X$ is a nilpotentlocally compact group equipped with an invariant geodesic metric.  Under the above hypotheses, we show that if $X$ issemi-locally-simply-connected, then it is a nilpotent Lie group equipped withan invariant Finsler or sub-Finsler metric, and for large enough $i$, there aresubgroups $\Lambda_i \leq \pi_1(X_i) $ with surjective morphisms $\Lambda_i\to\pi_1(X)$.

Limits of almost homogeneous spaces and their fundamental groups