Monte Carlo sampling is a powerful toolbox of algorithmic techniques widely used for a number of applications wherein some noisy quantity, or summary statistic thereof, is sought to be estimated. In this paper, we survey the literature for implementing Monte Carlo procedures using quantum circuits, focusing on the potential to obtain a quantum advantage in the computational speed of these procedures. We revisit the quantum algorithms that could replace classical Monte Carlo and then consider both the existing quantum algorithms and the potential quantum realizations that include adaptive enhancements as alternatives to the classical procedure.