  We investigate the properties of $\sim7000$ narrow-band selected galaxies
with strong H$\beta+$[OIII] and [OII] nebular emission lines from the High-$z$
Emission Line Survey (HiZELS) between $z \sim 0.8 - 5.0$. Our sample covers a
wide range in stellar mass ($M\sim10^{7.5 - 12.0}$ M$_\odot$), rest-frame
equivalent widths (EW$_\mathrm{rest}\sim 10 - 10^5$ \AA), and line luminosities
($L\sim10^{40.5 - 43.2}$ erg s$^{-1}$). We measure the H$\beta+$[OIII] and
[OII]-selected stellar mass functions out to $z \sim 3.5$ and find that both
$M_\star$ and $\phi_\star$ increases with cosmic time, which may be due to the
[OIII] selection including an increasing fraction of AGN at lower redshifts.
The [OII]-selected stellar mass functions show a constant
$M_\star\sim10^{11.6}$ M$_\odot$ and a strong, increasing evolution with cosmic
time in $\phi_\star$ in line with H$\alpha$ studies. We also investigate the
EW$_\mathrm{rest}$ evolution as a function of redshift with a fixed mass range
(10$^{9.5 - 10.0}$ M$_\odot$) and find an increasing trend best represented by
$(1+z)^{3.81\pm0.14}$ and $(1+z)^{2.72\pm0.19}$ up to $z\sim 2$ and $\sim 3$
for H$\beta$+[OIII] and [OII] emitters, respectively. This is the first time
that the EW evolution has been directly measured for H$\beta+$[OIII] and [OII]
emitters up to these redshifts. There is evidence for a slower evolution for $z
> 2$ in the H$\beta+$[OIII] EW and a decreasing trend for $z > 3$ in the [OII]
EW evolution, which would imply low [OII] EW at the highest redshifts and
higher [OIII]/[OII] line ratios. This suggests that the ionization parameter at
higher redshift may be significantly higher than the local Universe. Our
results set the stage for future near-IR space-based spectroscopic surveys to
test our extrapolated predictions and also produce $z > 5$ measurements to
constrain the high-$z$ end of the EW$_\mathrm{rest}$ and [OIII]/[OII]
evolution.
