The normal inverse Gaussian (NIG) process is a Lévy process with no Brownian component and NIG-distributed increments. The NIG process can be constructed either as a process with NIG increments or, alternatively, via random time change of Brownian motion using the inverse Gaussian process to determine time. The article presents the normal inverse Gaussian distribution function and the corresponding characteristic function and Lévy measure. Further, the NIG process is used to construct a market model for financial assets. Option pricing can be done using the NIG density function, the NIG Lévy characteristics, or the NIG characteristic function.