Binary response data is often modeled using the logistic regression model. Experimental design theory for the logistic model appears to be increasingly important as experimentation becomes more complex and expensive. The optimal design work is extremely valuable in areas such as biomedical and environmental applications. Most design research dealing with the logistic model has been concentrated on the one-variable case. Relative little has been done for the two-variable model. The primary goal of this research is to develop and study efficient and practical experimental design procedures for fitting the logistic model with two independent variables. Optimal designs are developed addressing D optimality, Q optimality, and the estimation of interaction between the design variables. The two-variable models with and without interaction usually have to be handled separately. The equivalence theory concerning D optimal designs is studied. The designs are compared using their relative efficiencies in the presence of interaction. Robustness to parameter misspecification is investigated. Bayesian design procedures are explored to provide relatively more robust experimental plans. / Ph. D.