Clearly written and well organized, this treatise presents material not found in other books on Lie groups. It systematically explores the structural aspects of complex Lie groups, beginning with general concepts then moving to the theory of representative functions of Lie groups and the extension problem of representations. This is followed by a discourse on complex analytic groups that carry the structure of affine algebraic groups compatible with their analytic group structure. The author then uses the results of his earlier discussions to determine the observability of subgroups of complex Lie groups.