  We give a direct proof of the magnetic Aharonov-Bohm effects without using
the scattering theory and the theory of inverse boundary value problems. This
proof can serve as a framework for a physical experiment to confirm the
magnetic AB effect. We prove also the electric AB effect and we suggest a
physical experiment to demonstrate the electric AB effect. In addition, we
consider a combined electric and magnetic AB effect and we propose a new
inverse problem for the time-dependent Schr\"odinger equations. Finally we
study the gravitational AB effect.
