  This Colloquium examines the field of the EPR Gedankenexperiment, from the
original paper of Einstein, Podolsky and Rosen, through to modern theoretical
proposals of how to realize both the continuous-variable and discrete versions
of the EPR paradox. We analyze the relationship with entanglement and Bell's
theorem, and summarize the progress to date towards experimental confirmation
of the EPR paradox, with a detailed treatment of the continuous-variable
paradox in laser-based experiments. Practical techniques covered include
continuous-wave parametric amplifier and optical fibre quantum soliton
experiments. We discuss current proposals for extending EPR experiments to
massive-particle systems, including spin-squeezing, atomic position entangle-
ment, and quadrature entanglement in ultra-cold atoms. Finally, we examine
applications of this technology to quantum key distribution, quantum
teleportation and entanglement-swapping.
