Conjoint Analysis (CA) is a mainstream method in market research studying the consumer behaviour. This method entails choice of experimental designs often dealing with large numbers of attributes and attribute levels. This thesis introduces the use of Partially Augmented Design (PAD) in CA and develops the universal optimality of two specific PAD's for use with large numbers of attribute levels. An example is presented to show the efficiency of the universally optimal PAD.  Uniform design, based on the work of Fang and Wang (1981), is introduced as an alternative to PAD in CA. A theorem previously developed by the author concerning the restriction in use of uniform design is presented. Optimal uniform designs are proposed. These designs are not orthogonal and hence they are also useful in the context of CA, because substantial correlations may exist between attributes.  The next important step in CA is proper statistical analysis. Appropriateness of random and mixed effects models, which are different from traditional fixed effects models used for analyzing CA data, is highlighted in several examples.