Abstract: The study of heirarchial structures distributed as independent multiplicative cascades can be traced back at least to the efforts by the Russian school led by Andrei N. Kolmogorov on the statistical theory of turbulence. However, largely owing to the intriguing statistical/ geometric scaling properties so heavily emphasized by Benoit Mandelbrot, and to the rich mathematical foundations begun by Jean-Pierre Kahane and Jacques Peyriere in the middle 1970's and continuing to the present, there has been a growing interest in both the mathematical theory of cascades and its applications to other characteristically highly variable and intermittant phenomena. In this talk we will present an overview of research in this area both from the points of view of the present status of the theory and outstanding mathematical and statistical problems. The talk is primarly based on the papers and references therein from the pulications listed below. The paper, Some Mathematical Aspects of Rainfall, Landforms, and Floods, is the most recent comprehensive survey of our work in this area. Selected Publications Holley, R., and E. Waymire (1992). Multifractal dimensions and scaling exponents for strongly bounded random cascades, Annals Appl. Prob., 2 (4), 819-845. Gupta, V. K., and E. C. Waymire (1993). A statistical analysis of mesoscale rainfall as a random cascade, J. Appl. Meteorol., 32(2), 251--267. Waymire, E.C. and S.C. Williams (1994). A general decomposition theory for random cascades Bull. Amer. Math. Soc. 31(2) 216-222. She, Z.S., E. Waymire (1995). Quantized energy cascade and log-Poisson statistics in fully developed turbulence, Phys. Rev. Letters 74(2), 262-265. Waymire, E. C., and S. C. Williams (1995). Multiplicative cascades: Dimension spectra and dependence, Jour. of Fourier Anal. and Appl. Special Issue in Honor of J-P Kahane 589-609. Waymire, E.C. and S.C. Williams (1996). A cascade decomposition theory with applications to Markov and exchangeable cascades, Trans. Amer. Math. Soc. 348(2), 585-632 . Waymire, E. C., and S. C. Williams (1997). Markov cascades, Classical and Modern Branching Processes, R.B. Athreya, P. Jagers eds, IMA Vol. 84,Springer-Verlag, 305-321 Ossiander, M., E. Waymire, Q. Zhang (1997) Some width function asymptotics for weighted trees", Ann. Appld. Probab. 3(7). V.K. Gupta and E. Waymire (1997,in press). Some mathematical aspects of rainfall, landforms, and floods in: Rainfall, Landforms, and Floods, O. Barndorff-Nielsen, V. Perez-Abreu, V.K. Gupta, E. Waymire eds, World Scientific Press.