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The critical behavior of two-dimensional ${\rm O}(N)$ $\sigma$ models with$N\leq 2$ on the square, triangular, and honeycomb lattices is investigated byan analysis of the strong-coupling expansion of the two-point fundamentalGreen's function $G(x)$, calculated up to 21st order on the square lattice,15th order on the triangular lattice, and 30th order on the honeycomb lattice.For $N<2$ the critical behavior is of power-law type, and the exponents$\gamma$ and $\nu$ extracted from our strong-coupling analysis confirm exactresults derived assuming universality with solvable solid-on-solid models. At$N=2$, i.e., for the 2-$d$ XY model, the results from all lattices consideredare consistent with the Kosterlitz-Thouless exponential approach tocriticality, characterized by an exponent $\sigma=1/2$, and with universality.The value $\sigma=1/2$ is confirmed within an uncertainty of few per cent. Theprediction $\eta=1/4$ is also roughly verified. For various values of $N\leq2$, we determine some ratios of amplitudes concerning the two-point function$G(x)$ in the critical limit of the symmetric phase. This analysis shows thatthe low-momentum behavior of $G(x)$ in the critical region is essentiallyGaussian at all values of $N\leq 2$. New exact results for the long-distancebehavior of $G(x)$ when $N=1$ (Ising model in the strong-coupling phase)confirm this statement.Comment: 33 pages incl. 1 fig., REVTeX+psfig, minor corrections to  bibliograph

Strong-coupling analysis of two-dimensional O(N) σ models withN⩽2 on square, triangular, and honeycomb lattices