1. Preliminaries.- I. O*-Algebras and Topologies.- 2. O-Families and Their Graph Topologies.- 3. Spaces of Linear Mappings Associated with O-Families and Their Topologization.- 4. Topologies for O-Famiiies with Metrizable Graph Topologies.- 5. Ultraweakly Continuous Linear Functionals and Duality Theory.- 6. The Generalized Calkin Algebra and the*-Algebra +( ).- 7. Commutants.- II:*-Representations.- 8. Basics of*-Representations.- 9. Self-Adjoint Representations of Commutative*-Algebras.- 10. Integrable Representations of Enveloping Algebras.- 11. n-Positivity and Complete Positivity of*-Representations.- 12. Integral Decompositions of*-Representations and States.- Symbol Index.