This book represents an important contribution to the analysis of elliptic partial differential equations on compact manifolds. For the sake of clearness the authors restrict themselves to the Yamabe equation and its generalizations, that includes the Schrödinger operator on the left-hand side and the critical (from the point of view of the embedding) nonlinearity on the right hand side. The structure of Sobolev solutions to such problems is known for more than twenty five years and is characterized as the sum of three items: the first part is a solution of the limit equation, the second part consists of a finite sum of bubbles, and the third term vanishes strongly in the Sobolev space. Within the book the authors develop a theory based on pointwise estimates that results in verifying the validity of the same asymptotic structure in the space of continuous functions. Before doing so, the authors provide the basic facts on Riemannian geometry and nonlinear analysis on manifolds, present the existence of Palais-Smale sequences and the existence of strong solutions of minimal and arbitrary energies for the Yamabe equation, and recall the above mentioned decomposition of Sobolev solutions. This all makes the book self-contained and despite the technicalities well accessible to the interested reader.