Banach function spaces form a class of Banach spaces consisting of functions defined on measure spaces. Prominent examples are Orlicz and Lorentz spaces. The first part of this chapter introduces the nonincreasing rearrangement of a random variable and the corresponding maximal function. It is shown that both are related by the Hardy-Littlewood maximal theorem. The maximal function is also related to the K-functional from interpolation theory. Interpolation of Banach spaces is a method from functional analysis which is used to construct a new Banach space as an intermediate space from two other Banach spaces. After having dealt with these basic relations, Banach function spaces are introduced and Lorentz and Orlicz spaces are discussed. The chapter concludes with a class of Lorentz and Orlicz spaces that describe different exponential tail behaviours of random variables.