Suppose Ω is a bounded open subset of ℝn with C2 boundary ∂Ω having nonnegative mean curvature. We examine the regularity at the boundary of solutions u to the minimal surface equation having boundary values ϕ. If ϕ has modulus of continuity β we give a modulus of continuity for u which depends on β and the behaviour of the mean curvature of ∂Ω. If ϕ is Lipschitz continuous then we show that u is Hölder continuous with some exponent α (explicitly obtained) that depends on the Lipschitz constant for ϕ. Finally we give examples showing the above results are best possible.