This paper introduces a new generalized polynomial dimensional decomposition (PDD) comprising measure-consistent multivariate orthogonal polynomials in dependent random variables. Unlike existing PDD, no tensor-product structure is assumed or required. Important mathematical properties of the generalized PDD are studied by constructing orthogonal decomposition of polynomial spaces, explaining completeness of orthogonal polynomials for prescribed assumptions, and demonstrating mean-square convergence to the correct limit. Analytical formulae are proposed to calculate the mean and variance of a truncated generalized PDD for a general output variable in terms of the expansion coefficients. Two numerical examples, the one derived from a stochastic boundary-value problem and the other entailing a random eigenvalue problem, illustrate the generalized PDD for second-moment error analysis and estimation of the probabilistic characteristics of eigensolutions.