We investigate a new method of extracting nucleon form factors by using measured asymmetries of polarized leptons scattered off polarized targets. Systematic errors can be eliminated that way, unlike the conventional analysis via Rosenbluth plots. The kinematics can be chosen such that the asymmetry receives contributions selectively from either GM2(q2) or GE(q2)GM(q2). The latter combination allows a better determination of GM(q2) for small q2. We also discuss lepton mass effects in dσ/dΩ. While such mass effects reduce the suppression of magnetic contributions to dσ/dΩ for small q2, they do not in general result in a greater sensitivity of dσ/dΩ or of the longitudinal asymmetries to a variation of the magnetic form factor, unless one chooses specific kinematics, as is demonstrated here. The transverse asymmetry, however, displays naturally a greater sensitivity to GM(q2) for the muon as compared to the electron. Such experiments are feasible at both the electron facilities (Bates, CEBAF, MAMI, and ELSA) and the muon facilities (SIN, TRIUMF, and LAMPF). Constituent models of the nucleon (quark confinement models) emphasize the low -q2 region of the form factors, where data presently are still incomplete. This has implications for the study of the nuclear response function in quasifree scattering (μ,μ’p) from nuclei using polarized muons.