The Josephson junction provides two strong nonlinearities, the Josephson inductance utilized in parametric amplification and the sharp resistive knee used in SIS mixing. Both of these nonlinearities can be strong compared to a characteristic quantum current or voltage scale. This opens the possibility of using Josephson junctions to carry out fundamental experiments in quantum electronics that would be difficult or impossible to carry out at optical frequencies due to the lack of a correspondingly large nonlinearity in optical media. Here we propose a number of such experiments ranging from the generation of squeezed states with Josephson-parametric amplifiers to the generation of quantum-mechanical superpositions of macroscopically distinguishable states via Josephson-transmission lines. How the properties of such states can be investigated using SIS mixers will also be described. Squeezed states may be technologically useful in sensitive and precision measurement. How such states can be used to enhance interferometer sensitivity or reduce noise in phase sensitive measurement will also be described.