  We consider first passage percolation (FPP) with passage times generated by a
general class of models with long-range correlations on $\mathbb{Z}^d$, $d\geq
2$, including discrete Gaussian free fields, Ginzburg-Landau $\nabla \phi$
interface models or random interlacements as prominent examples. We show that
the associated time constant is positive, the FPP distance is comparable to the
Euclidean distance, and we obtain a shape theorem. We also present two
applications for random conductance models (RCM) with possibly unbounded and
strongly correlated conductances. Namely, we obtain a Gaussian heat kernel
upper bound for RCMs with a general class of speed measures, and an exponential
decay estimate for the Green function of RCMs with random killing measures.
