Meshes approximate the boundaries of a geometry when the boundaries are curved. The accuracy of the mesh then affects the error of computations of initial boundary value problems for partial differential equations, especially when using high order methods. Here, we derive global estimates for the error in solutions of linear hyperbolic systems due to inaccurate boundary geometry. We show that the error is bounded by data and bounded in time when the solutions in the true and approximate domains are bounded. Just evaluating boundary data at the correct location has a secondary effect on the error, whereas the primary errors are from the Jacobian and metric terms. In two space dimensions, specifically, we show that to lowest order the errors are proportional to the errors in the boundary curve locations and their derivatives. Therefore, high order accuracy computations cannot be obtained unless the mesh is also high order. The results illustrate the importance of accurately approximating boundaries and should be helpful guides for high-order mesh generation for advection-dominated problems and the design of optimization algorithms for boundary approximations.