In the book, bifurcation problems for non-linear operator equations in infinite dimensional spaces are studied. To read the book requires certain knowledge, hence the first three chapters are devoted to a review (without proofs) of basic notions and facts from linear functional analysis, nonlinear functional analysis (e.g., the implicit function theory) and from the theory of analytic operators in Banach spaces. Main facts on holomorphic functions of several complex variables, real analytic functions of several real variables and on (finite-dimensional) analytic varieties, can be found in the next three chapters respectively. Analytic sets (in the complex setting, or their real version) have a nice, distinguished structure, which can be used in a study of analytic operator equations in infinite dimension. A tool needed for such a relation is a suitable version of the implicit function theorem, reducing infinite dimensional questions to finite dimension. The last two chapters contain applications of previous methods to steady periodic water waves.