Motivated by a variety of theories that predict new effects, we tested the gravitational 1/r^2 law at separations between 10.78 mm and 138 microns using two different 10-fold azimuthally symmetric torsion pendulums and rotating 10-fold symmetric attractors. Our work improves upon other experiments by up to a factor of about 100. We found no deviation from Newtonian physics at the 95% confidence level and interpret these results as constraints on extensions of the Standard Model that predict Yukawa or power-law forces. We set a constraint on the largest single extra dimension (assuming toroidal compactification and that one extra dimension is significantly larger than all the others) of R_* <= 160 microns, and on two equal-sized large extra dimensions of R_* <= 130 microns. Yukawa interactions with |alpha| >= 1 are ruled out at 95% confidence for lambda >= 197 microns. Extra-dimensions scenarios stabilized by radions are restricted to unification masses M_* >= 3.0 TeV/c^2, regardless of the number of large extra dimensions. We also provide new constraints on power-law potentials V(r) \propto r^{-k} with k between 2 and 5 and on the \gamma_5 couplings of pseudoscalars with m <= 10 meV/c^2.