Abstract: We will study asymptotic behavior of a simple random walk in random environments. As a driving example, we will look at percolation clusters of various sorts (including iid Bernoulli percolation as well as models carrying long-range correlations like random interlacements, its vacant set, the FK random cluster model, level sets of Gaussian free field etc.). We will sketch a method for proving an almost sure (i..e quenched) large deviation principle for the SRW and allude to the equivalence of this LD behavior to that of a homogenization of a random Hamilton-Jacobi PDE. Similar questions can be asked for the averaged (annealed) behavior of the SRW and time permitting, we will try to address a simple question: how much impact does the inherent impurity (disorder) of the environment have on the rate function(s), resp. on the effective homogenized equation(s)?