  Examples of operator algebras with involution include the operator
$*$-algebras occurring in noncommutative differential geometry studied recently
by Mesland, Kaad, Lesch, and others, several classical function algebras,
triangular matrix algebras, (complexifications) of real operator algebras, and
an operator algebraic version of the {\em complex symmetric} operators studied
by Garcia, Putinar, Wogen and others. We investigate the general theory of
involutive operator algebras, and give many applications.
