Quantum mechanics permits the description of thermal machines that operate continuously rather than in cycles, whose operation is characterized by work and heat currents at nonequilibrium steady-states. At this scale, current fluctuations can reach large magnitudes relative to their associated mean values, leading to new considerations, in addition to power and efficiency, for assessing performance. In this talk, I will introduce two basic models of continuous thermal machines--the thermoelectric engine and the quantum absorption refrigerator--for which techniques exist to calculate exact values of mean currents and fluctuations. I will then discuss a pair of novel bounds: the ratio of fluctuations in a continuous thermal machine's output current to those of its input current is bounded from below by the square of the machine's efficiency, and from above by the square of the relevant Carnot bound. These bounds have been proven for continuous thermal machines operating near equilibrium. They have also been shown to hold far from equilibrium in the limit of "tight-coupling", wherein a direct proportionality holds between work and heat currents. I will outline these results, as well as recent work on establishing the validity of these bounds for ensembles of tight-coupling systems.