The Department of Energy, Advanced Scientific Computing Research (DOE/ASCR) program held a four-day virtual workshop on "Randomized Algorithms for Scientific Computing (RASC)" on December 2-3 in 2020 and January 6-7 in 2021. The subsequent report articulates how randomized algorithms research is motivated by application needs and drivers such as: Massive data from experiments, observations, and simulations; Forward problems; Inverse problems; Applications with discrete structure; Experimental designs; Software and libraries for scientific computing; Emerging hardware; and Scientific machine learning. Relevant examples of randomized algorithms include stochastic gradient descent for training scientific machine learning models, compressed sensing, and differential privacy algorithms.