We report the first experiments and digital and analog simulations which demonstrate the existence of chaotic regions in the I-V curves of dc- and rf-current-biased Josephson junctions. These junctions were formed of crossed Pb strips and were shunted with Au resistors. Chaos appears as negatively going tails on the trailing edges of the rf-induced steps; these tails, which may be as large as 50% of the voltage step width, have not previously been reported. The parameters for the occurrence of these tails center at c=4, =/p=0.15, irf=Irf/Ic=1.04, Ic=3×105 A/m2 at 4.2 K. The thermal noise of the shunting resistor was emulated by a Gaussian spectrum. The presence of such noise dramatically alters the substeps, spikes, and bifurcations predicted for zero temperature. With only small amounts of noise, such complexities disappear, and are replaced by a smooth tail on the step accompanied by broadband noise. There is good agreement between the experiments on a real junction, simulations with a phase-locked loop, and numerical calculations with a digital computer. © 1985 The American Physical Society.