Polynomial operators are a natural generalization of linear operators. Equations in such operators are the linear space analog of ordinary polynomials in one or several variables over the fields of real or complex numbers. Such equations encompass a broad spectrum of applied problems including all linear equations. Often the polynomial nature of many nonlinear problems goes unrecognized by researchers. This is more likely due to the fact that polynomial operators - unlike polynomials in a single variable - have received little attention. Consequently this comprehensive presentation is needed benefiting those working in the field as well as those seeking information about specific results or techniques. Polynomial Operator Equations in Abstract Spaces and Applications - an outgrowth of fifteen years of the author's research work - presents new and traditional results about polynomial equations as well as analyzes current iterative methods for their numerical solution in various general space settings. Topics include:Special cases of nonlinear operator equationsSolution of polynomial operator equations of positive integer degree nResults on global existence theorems not related with contractionsGalois theoryPolynomial integral and polynomial differential equations appearing in radiative transfer heat transfer neutron transport electromechanical networks elasticity and other areasResults on the various Chandrasekhar equationsWeierstrass theoremMatrix representationsLagrange and Hermite interpolationBounds of polynomial equations in Banach space Banach algebra and Hilbert spaceThe materials discussed can be used for the following studiesAdvanced numerical analysisNumerical functional analysisFunctional analysisApproximation theoryIntegral and differential equation