Efficient adaptive algorithms for elliptic PDEs with random data. We present a novel adaptive algorithm implementing the stochastic Galerkin finite element method for numerical solution of elliptic PDE problems with correlated random data. The algorithm employs a hierarchical a posteriori error estimation strategy which also provides effective estimates of the error reduction for enhanced approximations. These error reduction indicators are used in the algorithm to perform a balanced adaptive refinement of spatial and parametric components of Galerkin approximations. The results of numerical tests demonstrating the efficiency of the algorithm for three representative PDEs with random coefficients are reported. The software used for numerical experiments is available online.