Representation Theory and C *-algebras is devoted to the representation theory of solvable Lie groups and the associated non-commutative harmonic analysis including the study of C*-algebras. It contains full proofs of long-standing problems in the theory, including several polynomial conjectures and primitive zero ideals descriptions. It provides an in-depth study of their structural properties, the classification of unitary representations using the orbit method, and the underlying algebraic and analytic frameworks. The book is most suitable for doctoral students, postdoctoral fellows, and researchers specializing in Lie theory, noncommutative geometry, functional analysis, operator algebras, and theoretical physics. Features -       Complete solutions to polynomial conjectures (Corwin-Greenleaf and Duflo) in both nilpotent and exponential Lie group settings -       Complete results to long-standing problems about intertwining operators and the structure of primitive ideals in the exponential setting -       Comprehensive analysis of Casimir elements based on a new approach and their role in several related problems -       Detailed study of C*-algebras of solvable Lie groups, via methods using the Fourier transform and the spectral analysis -       Rich examples and counterexamples, including Heisenberg, thread-like, and G6 groups -       Bridge between classical and modern methods in representation theory, with applications to harmonic analysis and mathematical physics