This paper studies robust variants of an extended model of the classical Heterogeneous Vehicle Routing Problem (HVRP), where a mixed fleet of vehicles with different capacities, availabilities, fixed costs and routing costs is used to serve customers with uncertain demand. This model includes, as special cases, all variants of the HVRP studied in the literature with fixed and unlimited fleet sizes, accessibility restrictions at customer locations, as well as multiple depots. Contrary to its deterministic counterpart, the goal of the robust HVRP is to determine a minimum-cost set of routes and fleet composition that remains feasible for all demand realizations from a pre-specified uncertainty set. To solve this problem, we develop robust versions of classical node- and edge-exchange neighborhoods that are commonly used in local search and establish that efficient evaluation of the local moves can be achieved for five popular classes of uncertainty sets. The proposed local search is then incorporated in a modular fashion within two metaheuristic algorithms to determine robust HVRP solutions. The quality of the metaheuristic solutions is quantified using an integer programming model that provides lower bounds on the optimal solution. An extensive computational study on literature benchmarks shows that the proposed methods allow us to obtain high quality robust solutions for different uncertainty sets and with minor additional effort compared to deterministic solutions.