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We make a high-precision Monte Carlo study of two-and three-dimensional self-avoiding walks (SAWs) of length up to 80000 steps, using the pivot algorithm and the Karp-Luby algorithm. We study the critical exponents and 2 4 as well as several universal amplitude ratios; in particular, we make an extremely sensitive test of the hyperscaling relation d = 2 4 . In two dimensions, we con rm the predicted exponent = 3=4 and the hyperscaling relation; we estimate the universal ratios hR 2 g i=hR 2 e i = 0:14026 0:00007, hR 2 m i=hR 2 e i = 0:43961 0:00034 and = 0:66296 0:00043 (68% con dence limits). In three dimensions, we estimate = 0:5877 0:0006 with a correctionto-scaling exponent 1 = 0:56 0:03 (subjective 68% con dence limits). This value for agrees excellently with the eld-theoretic renormalization-group prediction, but there is some discrepancy for 1 . Earlier Monte Carlo estimates of , which were 0:592, are now seen to be biased by corrections to scaling. We estimate the universal ratios hR 2 g i=hR 2 e i = 0:1599 0:0002 and = 0:2471 0:0003; since > 0, hyperscaling holds. The approach to is from above, contrary to the prediction of the two-parameter renormalizationgroup theory. We critically reexamine this theory, and explain where the error lies. 6 Here \good solvent" means that we work at any xed temperature strictly above the theta temperature for the given polymer-solvent pair. 7 In (1.3) we have assumed for simplicity that the hyperscaling relation d = 2 4 is valid.6

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