We review the theory, fabrication, and implementation of the Josephson bifurcation amplifier (JBA). At the core of the JBA is a nonlinear oscillator based on a reactively shunted Josephson junction. A weak input signal to the amplifier couples to the junction critical current I(0) and results in a dispersive shift in the resonator plasma frequency omega(p). This shift is enhanced by biasing the junction with a sufficiently strong microwave current I(rf) to access the nonlinear regime where omega(p) varies with I(rf). For a drive frequency omega(d) such that Omega=2Q(1-omega(d)/omega(p))>3, the oscillator enters the bistable regime where two nondissipative dynamical states O(L) and O(H), which differ in amplitude and phase, can exist. The sharp I(0) dependent transition from O(L) to O(H) forms the basis for a sensitive digital threshold amplifier. In the vicinity of the bistable regime (Omega<3), analog amplification of continuous signals is also possible. We present experimental data characterizing amplifier performance and discuss two specific applications--the readout of superconducting qubits (digital mode) and dispersive microwave magnetometry (analog mode).