The theory of operator algebras is generally considered over the field of complex numbers and in the complex Hilbert spaces. So it is a natural and interesting problem: How is the theory in the field of real numbers? Up to now, the theory of operator algebras over the field of real numbers has seemed not to be introduced systematically and sufficiently.The aim of this book is to set up the fundamentals of real operator algebras and to give a systematic discussion for real operator algebras. Since the treatment is from the beginning (real Banach and Hilbert spaces, real Banach algebras, real Banach ∗ algebras, real C∗-algebras and W∗-algebras, etc.), and some basic facts are given, one can get some results on real operator algebras easily.The book is also an introduction to real operator algebras, written in a self-contained manner. The reader needs just a general knowledge of Banach algebras and operator algebras.