  We describe modern variants of Monte Carlo methods for Uncertainty
Quantification (UQ) of the Neutron Transport Equation, when it is approximated
by the discrete ordinates method with diamond differencing. We focus on the
mono-energetic 1D slab geometry problem, with isotropic scattering, where the
cross-sections are log-normal correlated random fields of possibly low
regularity. The paper includes an outline of novel theoretical results on the
convergence of the discrete scheme, in the cases of both spatially variable and
random cross-sections. We also describe the theory and practice of algorithms
for quantifying the uncertainty of a linear functional of the scalar flux,
using Monte Carlo and quasi-Monte Carlo methods, and their multilevel variants.
A hybrid iterative/direct solver for computing each realisation of the
functional is also presented. Numerical experiments show the effectiveness of
the hybrid solver and the gains that are possible through quasi-Monte Carlo
sampling and multilevel variance reduction. For the multilevel quasi-Monte
Carlo method, we observe gains in the computational $\varepsilon$-cost of up to
2 orders of magnitude over the standard Monte Carlo method, and we explain this
theoretically. Experiments on problems with up to several thousand stochastic
dimensions are included.
