This book adds to the great body of research that extends back to A. Weil and E. P. Wigner on the unitary representations of locally compact groups and their characters, i.e. the interplay between classical group theory and modern analysis. The groups studied here are the connected Lie groups of general type (not necessarily nilpotent or semisimple). Final results reflect Kirillov's orbit method; in the case of groups that may be non-algebraic or non-type I, the method requires considerable sophistication. Methods used range from deep functional analysis (the theory of $C^*$-algebras, factors from F. J. Murray and J. von Neumann, and measure theory) to differential geometry (Lie groups and Hamiltonian actions). This book presents for the first time a systematic and concise compilation of proofs previously dispersed throughout the literature. The result is an impressive example of the deepness of Pukanszky's work.