This volume presents a general theory for the approximation of regular and bifurcation branches of solutions of nonlinear equations. After recalling basic results of nonlinear approximation theory, the authors give the methodology for extending these regular branches on nonlinear problems, then proceed to discuss more complex singularities. The error estimates obtained are optimal and while they are described in terms of classical finite elements. The formalism is more general and can deal with mixed or hybrid finite elements as well as with spectral methods. The book is addressed to researchers and mathematical engineers concerned with approximation for elliptic nonlinear problems and is accessible to graduate students, requiring only basic knowledge of the subject.