This chapter introduces the basic elements of risk and financial assets pricing. Asset pricing is considered in two essential situations, complete and incomplete markets, and the definition and use of a number of essential financial instruments is described. Specifically, stocks (as underlying processes), bonds and derivative products (and in particular call and put European and American options) are considered. The intent of the chapter is neither to cover all the many techniques and approaches that are used in asset pricing, nor to provide a complete introduction to financial asset pricing and financial engineering. Rather, the intent of the chapter is to outline through applications and problems the essential mathematical techniques and financial economic concepts used to assess the value of risky assets. An extensive set of references is also included to direct the motivated reader to further and extensive research in this broad and evolving domain of economic and financial engineering and mathematics that deals with asset pricing. The first part of the chapter (The Introduction and Sect. 47.1) deals with a definition of risk and outlines the basic terminology used in asset pricing. Further, some essential elements of the Arrow–Debreu framework that underlies the fundamental economic approach to asset pricing are introduced. A second part (Sect. 47.2), develops the concepts of risk-neutral pricing, no arbitrage and complete markets. A number of examples are used to demonstrate how we can determine a probability measure to which risk-neutral pricing can be applied to value assets when markets are complete. In this section, a distinction between complete and incomplete markets is also introduced. Sections 47.3, 47.4 and 47.5 provide an introduction to and examples of basic financial approaches and instruments. First, Sect. 47.3, outlines the basic elements of the consumption capital asset-pricing model (with the CAPM stated as a special case). Section 47.4 introduces the basic elements of net present value and bonds, calculating the yield curve as well as the term structure of interest rates and provides a brief discussion of default and rated bonds. Section 47.5 is a traditional approach to pricing of options using the risk-neutral approach (for complete markets). European and American options are considered and priced by using a number of examples. The Black–Scholes model is introduced and solved, and extensions to option pricing with stochastic volatility, underlying stock prices with jumps as well as options on bonds are introduced and solved for specific examples. The last section of the chapter focuses on incomplete markets and an outline of techniques that are used in pricing assets when markets are incomplete. In particular, the following problems are considered: the pricing of rated bonds (whether they are default-prone or not), engineered risk-neutral pricing (based on data regarding options or other derivatives) and finally we also introduce the maximum-entropy approach for calculating an approximate risk-neutral distribution.