This book is based on the CBMS Regional Conference held at the University of California, Irvine, in July 1985. Jordan algebras have found interesting applications in seemingly unrelated areas of mathematics such as operator theory, the foundations of quantum mechanics, complex analysis in finite and infinite dimensions, and harmonic analysis on homogeneous spaces. The author describes some relevant results and puts them in a general framework, based on the concepts of $JB$-algebra and $JB^ast$-triple.