Monte Carlo (MC) and Quasi-Monte Carlo (QMC) methods have become important ways to solve complex integration problems in engineering, banking, and science computing.  MC uses random sampling to get better convergence, but QMC uses organized low-discrepancy patterns to do the same thing. This work looks at the theoretical bases, error limits, and real-world uses of both methods, with a focus on variance reduction, importance sampling, and mixed randomized QMC strategies. This paper discuss about these methods side by side shows how they combine processing complexity, accuracy, and stability, which is why they are so important in current applied mathematics and computational science.