Let X be a compact normal complex space, L be a big holomorphic line bundle on X and h be a continuous Hermitian metric on L. We consider the spaces of holomorphic sections H0(X, L⊗p) endowed with the inner product induced by h⊗p and a volume form on X, and prove that the corresponding sequence of normalized Fubini- Study currents converge weakly to the curvature current c1(L, heq) of the equilibrium metric heq associated to h. We also show that the normalized currents of integration along the zero divisors of random sequences of holomorphic sections converge almost surely to c1(L,heq), for very general classes of probability measures on H0(X,L⊗p). This is joint work with Turgay Bayraktar, George Marinescu, and Viêt-Anh Nguyên.