This text describes how fractal phemena, both deterministic and random, change over time, using the fractional calculus. The intent is to identify those characteristics of complex physical phemena that require fractional derivatives or fractional integrals to describe how the process changes over time. The discussion emphasizes the properties of physical phemena whose evolution is best described using the fractional calculus, such as systems with long-range spatial interactions or long-time memory. In many cases, classic analytic function theory cant serve for modeling complex phemena; Fractal Operators shows how classes of less familiar functions, such as fractals, can serve as useful models in such cases. Because fractal functions, such as the Weierstrass function (long kwn t to have a derivative), do in fact have fractional derivatives, they can be cast as solutions to fractional differential equations. The traditional techniques for solving differential equations, including Fourier and Laplace transforms as well as Green's functions, can be generalized to fractional derivatives. Fractal Operators addresses a general strategy for understanding wave propagation through random media, the nlinear response of complex materials, and the fluctuations of various forms of transport in heterogeneous materials. This strategy builds on traditional approaches and explains why the historical techniques fail as phemena become more and more complicated.