The main interest of my research is in distribution theory, which lies in the center of probability and statistics. It has been a challenging topic in distribution theory during last several decades to obtain probability density and distribution functions and related probabilistic quantities of complicated functions of random variables. The main approximation methods beside obtaining exact functions are Monte Carlo simulation, numerical inversion, higher order asymptotic approximation, and matrix analytic distributions. In this talk, existing higher order asymptotic approximation methods such as Edgeworth expansion and Saddlepoint approximation are discussed and three new moment based approximation methods are introduced. Their performances are compared with those of the existing methods, and several examples show that the new moment based methods can outperform the existing techniques. The three new methods are (1) Polynomially Adjusted Approximant (PAM), (2) Di erentiated LogDensity Approximant (DLA), (3) Pade-based Transform Approximation Method (PTAM). The new methods are applied for Delta-Gamma method in nance and VaR and TVaR of aggregate losses in actuarial science.