Orthogonal designs of special type have been extensively studied, and it is the existence of these special types that has motivated our study of the general problem of the existence of orthogonal designs.
This paper is organized in the following way. In the first section we give some easily obtainable necessary conditions for the existence of orthogonal designs of various order and type. In Section 2 we briefly survey the examples of such designs that we have found in the literature. In the third section we describe several methods for constructing orthogonal designs. In the fourth section we obtain some sharper necessary conditions for the existence of orthogonal designs. In the fifth section we apply the results obtained to calculate designs of small order and also improve some of the results of. We conclude this paper with a collection of open questions and conjectures.