In this volume the author studies noncommutative domains $\mathcal{D}_f\subset B(\mathcal{H})^n$ generated by positive regular free holomorphic functions $f$ on $B(\mathcal{H})^n$, where $B(\mathcal{H})$ is the algebra of all bounded linear operators on a Hilbert space $\mathcal{H}$. Table of Contents: Introduction; Operator algebras associated with noncommutative domains; Free holomorphic functions on noncommutative domains; Model theory and unitary invariants on noncommutative domains; Commutant lifting and applications; Bibliography. (MEMO/205/964).