  We present a detailed study of the structural, electronic, magnetic and
ferroelectric properties of prototypical \textit{proper} and \textit{improper}
multiferroic (MF) systems such as BiFeO$_{3}$ and orthorhombic HoMnO$_{3}$,
respectively, within density functional theory (DFT) and using the
Heyd-Scuseria-Ernzerhof hybrid functional (HSE). By comparing our results with
available experimental data as well as with state-of-the-art GW calculations,
we show that the HSE formalism is able to account well for the relevant
properties of these compounds and it emerges as an accurate tool for predictive
first-principles investigations on multiferroic systems. We show that effects
beyond local and semilocal DFT approaches (as provided by HSE) are necessary
for a realistic description of MFs. For the electric polarization, a decrease
is found for MFs with magnetically-induced ferroelectricity, such as HoMnO$_3$,
where the calculated polarization changes from $\sim$ 6 $\mu C/cm^2$ using
Perdew-Burke-Ernzerhof (PBE) to $\sim$ 2 $\mu C/cm^2$ using HSE. However, for
\textit{proper} MFs, such as BiFeO$_{3}$, the polarization slightly increases
upon introduction of exact exchange. Our findings therefore suggest that a
general trend for the HSE correction to bare density functional cannot be
extracted; rather, a specific investigation has to be carried out on each
compound.
