A crossover design is of use, because it can often improve the efficiency of the parallel group and other commonly-used designs with repeated measurements in clinical trials. The goals of dissertation are divided into three components: (1) improving efficiency of existing test procedures for comparing three treatments under a three-period crossover design; (2) studying the robustness of test procedure and treatment effect estimator based on the maximum likelihood estimator to the normal assumption for random subject effects for AB/BA design; and (3) studying the impact of carryover on the performance of test procedures, point and interval estimators for the treatment effect under various crossover designs. Three research goals are summarized in the following: Goal 1 Based on the idea of Prescotts test accounting for untied responses, two exact tests and an asymptotic test are developed for the three-period crossover trial comparing three-treatment. Monte Carlo simulation is used to evaluate the performance of these procedures and compare power of this procedure with the test procedures developed elsewhere with accounting for only discordant responses. The data taken from a trial of comparing two different doses of an analgesic with placebo for the relief of primary dysmenorrhea are used to illustrate the use of test procedures developed here. Goal 2 When subject responses are dichotomous under a simple crossover (or AB/BA) design, the normal random effects logistic regression model is commonly assumed. However, the normality for the random effects is unlikely to hold. The investigation on the impact due to this misspecification of distribution for random effects on hypothesis testing and estimation based on the maximum likelihood estimator (MLE) is of practical importance and interest. Based on Monte Carlo simulation, Type I error for testing equality of treatments is found to be minimally affected by misspecifying the random effects as the normal distribution. Furthermore, when the variation of responses between subjects is small and the number of subjects per group is large, the influence due to such misspecification on power, bias and mean-squared-error (MSE) for point estimation, as well as the coverage probability and the average length for interval estimation is generally minimal. However, this influence can be substantial when the variation of responses between subjects is large and the number of subjects per group is small. Thus, calculation of the required sample size with assuming the normal random effects can be inaccurate. The data taken from the trial comparing a drug with placebo in treating cerebrovascular deficiency are employed to illustrate the potential difference in inference between various random effects distributions. Goal 3 Carry-over effects can confound the estimation of treatment effects in crossover trials. One common approach is to adopt a model, including parameters for carry-over. However, the inclusion of these parameters for carry-over can compromise the efficiency. Moreover, if the assumed model does not adequately adjust carry-over, the treatment effect estimators can be still biased. The study of efficiency and accuracy for various carry-over adjustments under a variety of designs, including the AB/BA design, the incomplete block design and complete block design for comparing three treatments, is considered. As what one will expect, the assumed model with an increase in the number of parameters for carry-over may reduce the bias, but can cause the loss of efficiency. We note further that the main advantage and motivation for using a crossover design instead of the parallel group design may completely disappear especially when the carry-over model structure is complicated. The data taken from the trial comparing a drug with placebo in treating cerebrovascular deficiency and the data taken from a trial of comparing two difference doses of an analgesic with placebo for the relief of primary dysmenorrhea is used to illustrate the carry-over impact on the efficiency.