This paper establishes a precise high-dimensional asymptotic theory for Boosting on separable data, taking statistical and computational perspectives. We consider the setting where the number of features (weak learners) $p$ scales with the sample size $n$, in an over-parametrized regime. On the statistical front, we provide an exact analysis of the generalization error of Boosting, when the algorithm interpolates the training data and maximizes an empirical $L_1$ margin. The angle between the Boosting solution and the ground truth is characterized explicitly. On the computational front, we provide a sharp analysis of the stopping time when Boosting approximately maximizes the empirical $L_1$ margin. Furthermore we discover that, the larger the margin, the smaller the proportion of active features (with zero initialization). At the heart of our theory lies a detailed study of the maximum $L_1$ margin, using tools from convex geometry. The maximum $L_1$ margin can be precisely described by a new system of non-linear equations, which we study using a novel uniform deviation argument. Preliminary numerical results are presented to demonstrate the accuracy of our theory.