The object of this thesis is to study the classical Heath-Jarrow-Morton(HJM) model for interest rates, and the corresponding London Interbank Offered Rate(LIBOR) model, when the noise is driven by an It\^o-L\'evy process instead of only a Brownian motion. When the model is driven by only a Brownian motion we have well known theory concerning the risk neutral measures and how to compute arbitrage free prices for options. In this thesis we will find corresponding results when the market is modeled by jump diffusions. One of the problems with markets modeled by jump diffusions is that these models will in general be incomplete, so we will get several equivalent local martingale measures(ELMM), so one of the problems we will look at is how to find such measures. Next we will look at how to compute the price of a European call option for a general ELMM, this will be done with the use of Fourier transforms and computation of a characteristic function. At last we will look at a utility maximization problem and how to find investment strategies for this problem, and one of the methods we will use to find this is a duality method.