Hadronic matrix elements of neutral-meson mixing through lattice QCD

Neutral-meson mixing is loop suppressed in the Standard Model, leading to the possibility of enhanced sensitivity to new physics. The uncertainty in Standard Model predictions for $B$-meson oscillation frequencies is dominated by theoretical uncertainties within the short-distance $B$-meson hadronic matrix elements, motivating the need for improved precision. In $D$-meson mixing, the Standard Model short-distance contributions are further suppressed by the GIM mechanism allowing for the possibility of large new physics enhancements. A first-principle determination of the $D$-meson short-distance hadronic matrix elements will allow for model-discrimination between the new physics theories. I review recently published and ongoing lattice calculations of hadronic matrix elements in $B$ and $D$-meson mixing with emphasis on the Fermilab lattice and MILC collaboration effort on the determination of the $B$ and $D$-meson mixing hadronic matrix elements using the methods of lattice QCD.