Boundary value problems and smoothness of solutions to nonlinear elliptic equations and linear degenerate elliptic equations have been studied for more than a century. However, the study of nonlinear degenerate elliptic equations is not enough taken up. This book is based on research of the author on semilinear degenerate elliptic equations, i. e. nonlinear equations the principal parts of which are linear and degenerate elliptic. The first and second parts, after an introduction, cover the following topics: infinite differentiability, analyticity, Gevrey regularity of solutions. The third part is devoted to the study of the boundary value problems. Here the phenomena of critical exponents are discussed. This book will be a useful reference source for graduate students and researchers in differential equations, complex analysis, as well as in nonlinear analysis.