This book explores some basic roles of Lie groups in linear analysis, with particular emphasis on the generalizations of the Fourier transform and the study of partial differential equations. It is a valuable resource for both graduate students and faculty, and requires only a background in Fourier analysis and basic functional analysis, plus the first few chapters of a standard text on Lie groups. The basic method of noncommutative harmonic analysis, a generalization of Fourier analysis, is to synthesize operators on a space on which a Lie group acts from operators on an irreducible representation of the Lie group. Though the general study is far from complete, this book covers a great deal of the progress that has been made on important classes of Lie groups. Also, while the author does treat some general approaches, particularly for compact and nilpotent groups, the emphasis is on particular groups. Unlike many other books on harmonic analysis, this book focuses on the relationship between harmonic analysis and partial differential equations. The author considers many classical PDEs, particularly boundary value problems for domains with simple shapes, that exhibit noncommutative groups of symmetries. The book also contains detailed work on the harmonic analysis of the Heisenberg group and harmonic analysis on cones.