Data assimilation is a procedure for combining observations and forecasts of a system into a single, improved description of the system state. Because observations and forecasts are uncertain, they are each best described by probability distributions. The problem of combining these two distributions into a new, updated distribution that summarizes all our knowledge is solved by Bayes theorem. If the distributions are Gaussian, then the parameters of the updated distribution can be written as an explicit function of the parameters of the observation and forecast distributions. The assumption of Gaussian distributions is tantamount to assuming linear models for observations and state dynamics. The purpose of this chapter is to provide an introduction to the essence of data assimilation. Accordingly, this chapter discusses the data assimilation problem for Gaussian distributions in which the solution from Bayes theorem can be derived analytically. Practical data assimilation usually requires modifications of this assimilation procedure, a special case of which is discussed in the next chapter.