This chapter focuses on providing some technical background needed for the analysis on the surplus models in later chapters. This background is not meant to be comprehensive, i.e. well-known mathematical topics are normally assumed to be known and are not discussed in much detail. Rather, less commonly known topics specific to this monograph are discussed. A brief review is provided of Lagrange polynomials due to their importance in numerous subsequent places in the monograph. Then, the so-called ‘Dickson-Hipp’ operator, which generalizes both Laplace transforms and distribution tails and is of central importance in many of the models, is discussed. As much of the monograph utilizes defective renewal equation methodology, this topic and the closely related compound geometric and compound geometric convolution methodology are also reviewed. Finally, the important classes of mixed Erlang and Coxian distributions, which have attracted much attention in recent years in the applied probability and actuarial literature due to their mathematical tractability, are briefly summarized.