Field emitter tips can now be fabricated with radii of curvature of the order of nm or even the size of a single atom. To include these geometric effects, we have calculated the field emission tunneling currents for hyperboloidal and conical free-electron tip models using geometry-dependent image interactions and bias fields. The numerical results can be fitted by an equation of the form J=AV2 exp(-B/V-C/V2), where A, B, and C are constants depending on material and geometric parameters. The calculated results, plotted as log J/V2 vs 1/V, do not exhibit the straight line behavior predicted by the Fowler-Nordheim model for field emission from a planar surface. Furthermore, the calculated current densities are dramatically enhanced for both the hyperboloidal (rt=10 nm) and conical (cone half-angle=70°) emitter models. In addition, field emission energy distributions for both models are significantly different from that of the Fowler-Nordheim planar model.