  In this paper we introduce two new classes of operators that we call strongly
order continuous and strongly $\sigma$-order continuous operators. An operator
$T:E\rightarrow F$ between two Riesz spaces is said to be strongly order
continuous (resp. strongly $\sigma$-order continuous), if $x _\alpha
\xrightarrow{uo}0$ (resp. $x _n \xrightarrow{uo}0$) in $E$ implies $Tx _\alpha
\xrightarrow{o}0$ (resp. $Tx _n \xrightarrow{o}0$) in $F$. We give some
conditions under which order continuity will be equivalent to strongly order
continuity of operators on Riesz spaces. We show that the collection of all
$so$-continuous linear functionals on a Riesz space $E$ is a band of $E^\sim$.
