  This monograph resolves - in a dense class of cases - several open problems
concerning geodesics in i.i.d. first-passage percolation on $\mathbb{Z}^d$. Our
primary interest is in the empirical measures of edge-weights observed along
geodesics from $0$ to $n\xi$, where $\xi$ is a fixed unit vector. For various
dense families of edge-weight distributions, we prove that these empirical
measures converge weakly to a deterministic limit as $n\to\infty$, answering a
question of Hoffman. These families include arbitrarily small
$L^\infty$-perturbations of any given distribution, almost every finitely
supported distribution, uncountable collections of continuous distributions,
and certain discrete distributions whose atoms can have any prescribed sequence
of probabilities. Moreover, the constructions are explicit enough to guarantee
examples possessing certain features, for instance: both continuous and
discrete distributions whose support is all of $[0,\infty)$, and distributions
given by a density function that is $k$-times differentiable. All results also
hold for $\xi$-directed infinite geodesics. In comparison, we show that if
$\mathbb{Z}^d$ is replaced by the infinite $d$-ary tree, then any distribution
for the weights admits a unique limiting empirical measure along geodesics. In
both the lattice and tree cases, our methodology is driven by a new variational
formula for the time constant, which requires no assumptions on the edge-weight
distribution. Incidentally, this variational approach also allows us to obtain
new convergence results for geodesic lengths, which have been unimproved in the
subcritical regime since the seminal 1965 manuscript of Hammersley and Welsh.
