In general, diffusions cannot be directly simulated because we hardly know enough about their probability distributions. Discretization methods can be usually used for simulating the diffusions, but introduce bias. An exact simulation technique for one-dimensional diffusions has recently been proposed by Beskos and Roberts  based on the acceptance-rejection method. Their method completely eliminates the bias from discretization. We propose, in this dissertation, two simulation techniques based on the Beskos-Roberts method. First, we propose unbiased Monte Carlo estimators of the Greeks by taking advantages of the Beskos-Roberts method. Some exiting methods of the Greeks provide unbiased estimators theoretically. However, their implementation still requires discretization to estimate, which causes bias inevitably. The Beskos-Roberts method can overcome such difficulty. Second, we propose an approach to improve the plain Beskos-Roberts method. Under certain scenarios, the plain method can become inefficient because of small acceptance probabilities. To improve the acceptance probabilities, we suggest a new method by adapting the localization idea of Chen and Hunag .