Reduced-basis (RB) methods enable repeated and rapid evaluation of parametrized partial differential equation (PDE)-constrained input-output relationships required in the context of parameter estimation, design, optimization, and control. These methods have been successfully applied to problems with few parameters [O(3)]. Here we introduce efficient sampling algorithms that enable the efficient exploration of many parameters. We apply the RB methods to an illustrative heat conduction problem with P = 25 parameters, obtaining accurate and certified results in real time with significant computational savings relative to standard finite-element techniques.