We study the continuity of pullback and uniform attractors for non-autonomous dynamical systems with respect to perturbations of a parameter. Consider a family of dynamical systems parameterised by a complete metric space $��$ such that for each $��\in��$ there exists a unique pullback attractor $\mathcal A_��(t)$. Using the theory of Baire category we show under natural conditions that there exists a residual set $��_*\subseteq��$ such that for every $t\in\mathbb R$ the function $��\mapsto\mathcal A_��(t)$ is continuous at each $��\in��_*$ with respect to the Hausdorff metric. Similarly, given a family of uniform attractors $\mathbb A_��$, there is a residual set at which the map $��\mapsto\mathbb A_��$ is continuous. We also introduce notions of equi-attraction suitable for pullback and uniform attractors and then show when $��$ is compact that the continuity of pullback attractors and uniform attractors with respect to $��$ is equivalent to pullback equi-attraction and, respectively, uniform equi-attraction. These abstract results are then illustrated in the context of the Lorenz equations and the two-dimensional Navier-Stokes equations.