In this project we review the developments of several variations of the up-and-down design utilized in Phase I clinical trials to estimate the maximum tolerated dose (MTD) of a drug which corresponds to a fixed probability of response or the pre-specified toxicity rate in the target population. In these designs selection of dose levels is restricted to one level higher, one level lower or the same. Several methods of estimation of the MTD are investigated. Some comparison of the designs by Monte Carlo simulation are carried out by the quality of the estimator of the target dose using the isotonic estimator. The designs are investigated under the generalized logistic (for different values of the power) and the gamma distributions. The NR is found to perform best on the basis of the quality of estimator under these distributions. The BCD is found to perform best on the basis of the average proportion of toxicity for a pre-specified toxicity rate of 0.2 whereas the KROW performs best for a toxicity rate of 0.3. / Thesis / Master of Science (MS)