We develop a unified mathematical framework extending classical moment theory from discrete integer orders to a continuous spectrum of real orders f>0, providing systematic statistical characterization of complex systems exhibiting power-law behavior. This fractional moment theory addresses the fundamental problem in anomalous transport where traditional integer moments diverge for heavy-tailed distributions characteristic of Lévy flights, continuous time random walks, and chaotic advection. Through rigorous analysis of space-time fractional diffusion equations with Hilfer-composite time derivatives and Riesz-Feller space derivatives, we establish the operator-moment correspondence theorem proving that moments ⟨ | x | f ⟩ converge if and only if f, where α is the Lévy stability index governing asymptotic tail behavior u ( x ) ∼ | x | − ( 1 + α ) . We derive from first principles the universal scaling law ⟨ | x | f ⟩ = A f K µ , α f/α t µf/α with explicit coefficient formulas expressed through Gamma functions, establishing connections to Fox H-functions, Mittag-Leffler relaxation, and Wright functions. Complete proofs are provided using multiple independent methods including self-similarity analysis, Mellin transform techniques, and asymptotic expansions. Applications are developed for turbulent dispersion obeying Richardson’s four-thirds law, Lagrangian chaos characterized by finite-scale Lyapunov exponents, anomalous diffusion on fractal substrates, multifractal cascades, relaxation dynamics in glassy systems, epidemic spreading on scale-free networks, and extreme value distributions. The continuous parameter f enables extraction of scaling exponents and transport coefficients from systems where variance-based analysis fails entirely.