An operator algebra is a closed algebra of bounded operators on a Hilbert space. In operator theory and in the theory of selfadjoint operator algebras (C*-algebras and von Neumann algebras), it is hard to overestimate the importance of the role played by positive elements and their roots. With Charles Read we have introduced and studied a new notion of positivity for operator algebras, valid even in algebras with no nonzero positive elements in the usual sense, with an eye to extending certain C*-algebraic results and theories to more general algebras. We discuss this notion of positivity, including some results joint with Bearden and Sharma, and give some typical applications.