First-passage percolation is a random growth model on the cubic lattice Z^d. It models the spread of fluid in a random porous medium. This talk is about the asymptotic behavior of the first-passage time T(x), which represents the time it takes for a fluid particle released at the origin to reach a point x on the lattice. The first part of the talk is about a new formula for the first-order behavior of T(x) as x goes to ∞, which results from a connection between first-passage percolation and the stochastic homogenization theory of Hamilton-Jacobi equations. The second part of the talk is about the connection between the second-order behavior of T(x) and random matrix theory, and our results towards to the so-called universality conjecture.