This monograph presents the first detailed exposition of the formal theory of Branching Space-Times (BST). The theory presented here by Nuel Belnap, Thomas Muller, and Tomasz Placek describes how real possibilities can be anchored in a spatio-temporal world that is rudimentarily relativistic. Things that are possible in Cleveland are not possible in San Francisco; other things were possible in 1988 but are not possible in 2021. BST represents such possibilities using a mathematically rigorous framework that combines modality and rudimentary relativistic space-times. The book is divided into two parts: the first contains the exposition of the theory, including detailed proofs; the second contains three applications of BST in metaphysics and philosophy of science, focusing on the use of BST to represent pertinent forms of indeterminism in each area. <p/><em>This is an open access title available under the terms of a CC BY-NC-ND 4.0 International license. It is free to read at Oxford Scholarship Online and offered as a free PDF download from OUP and selected open access locations.</em>