A comprehensive overview of weak convergence of shastic processes and its application to the study of financial markets. Split into three parts, the first recalls the mathematics of shastic processes and shastic calculus with special emphasis on contiguity properties and weak convergence of shastic integrals. The second part is devoted to the analysis of financial theory from the convergence point of view. The main problems such as portfolio optimization, option pricing and hedging are examined, especially when considering discrete-time approximations of continuous-time dynamics. The third part deals with lattice- and tree-based computational procedures for option pricing both on sks and shastic bonds. More general discrete approximations are also introduced and detailed.