  The dispersion of a passive scalar in a fluid through the combined action of
advection and molecular diffusion is often described as a diffusive process,
with an effective diffusivity that is enhanced compared to the molecular value.
However, this description fails to capture the tails of the scalar
concentration distribution in initial-value problems. To remedy this, we
develop a large-deviation theory of scalar dispersion that provides an
approximation to the scalar concentration valid at much larger distances away
from the centre of mass, specifically distances that are $O(t)$ rather than
$O(t^{1/2})$, where $t \gg 1$ is the time from the scalar release. The theory
centres on the calculation of a rate function obtained by solving a
one-parameter family of eigenvalue problems which we derive using two
alternative approaches, one asymptotic, the other probabilistic. We emphasise
the connection between large deviations and homogenisation: a perturbative
solution of the eigenvalue problems reduces at leading order to the cell
problem of homogenisation theory. We consider two classes of flows in some
detail: shear flows and cellular flows. In both cases, large deviation
generalises classical results on effective diffusivity and captures new
phenomena relevant to the tails of the scalar distribution. These include
approximately finite dispersion speeds arising at large P\'eclet number
$\mathrm{Pe}$ (corresponding to small molecular diffusivity) and, for
two-dimensional cellular flows, anisotropic dispersion. Explicit asymptotic
results are obtained for shear flows in the limit of large $\mathrm{Pe}$. (A
companion paper, Part II, is devoted to the large-$\mathrm{Pe}$ asymptotic
treatment of cellular flows.) The predictions of large-deviation theory are
compared with Monte Carlo simulations that estimate the tails of concentration
accurately using importance sampling.
