Random variables with values in a Banach space generalize \(\mathbb {R}^d\) -valued random variables and have their fixed place in probability theory and applications. First Pettis’ measurability theorem about the equivalence of certain measurability concepts is shown. Then the Bochner, Pettis, and Dunford integrals are introduced, which generalize the Lebesgue integral to different degrees. After that, the strong law of large numbers and the Ito-Nisio theorem for Banach space valued random variables are shown. The chapter concludes with the construction of the Brownian motion as a Banach space valued random variable using the results of Ciesielski about Hölder functions. Here Banach function spaces are used to describe the modulus of continuity.