The book, Orthogonal Designs: Quadratic Forms and Hadamard Matrices, Marcel Dekker, New York-Basel, 1979, by A. V. Geramita and Jennifer Seberry, has now been out of print for almost two decades. Many of the results on weighing matrices presented therein have been greatly improved. Here we review the theory, restate some results which are no longer available and expand on the existence of many new weighing matrices and orthogonal designs of order 2n where n is odd. We give a number of new constructions for orthogonal designs. Then using number theory, linear algebra and computer searches we find new weighing matrices and orthogonal designs. We have reviewed completely the weighing matrix conjecture for orders 2n, n ≤ 35, n odd. The previously known results for weighing matrices for n ≤ 21 are summarized here, and new result given, leaving 3 unresolved cases. The results for weighing matrices for n ≥ 23 are presented here for the first time. For orders n, 23 ≤ n ≤ 25, 3 remain unsolved as do a further 106 cases for orders 27 ≤ n ≤ 49. We also review completely the orthogonal design conjecture for two variables in orders ≡ 2(mod 4). The results for orders 2n, n odd, 15 ≤ n ≤ 33 being given here for the first time.