This thesis presents an overview of the methodology for pricing contingent claims on the term structure of interest rates. Given an initial forward rate curve and a mechanism which describes how it fluctuates, we develop an arbitrage pricing model which yields contingent claim valuations which do not explicitly depend on the market prices for risk. In the first chapter, we formulated the basic postulates of the HJM approach and we derived the formula for the dynamics of the bond price. In the second chapter, we present two sufficient conditions for the absence of arbitrage for all bonds of different maturities, and we conclude that these two conditions are equivalent. In the third chapter, we assumed that the volatility σ of the forward rate is a deterministic function and we showed that the market prices for risk are derived from arbitrage values of interest rate-sensitive derivatives. In the fourth chapter, we obtained a closed-form expression for the arbitrage price of a European bond option under the assumption that volatilities are bounded. We also presented an explicit formula for the arbitrage price of a call option on a stock and proved that the formula for the put-call parity is valid.