An operator $T $ from a vector lattice $E$ into a normed lattice $F$ is called unbounded $\sigma$-order-to-norm continuous whenever $x_{n}\xrightarrow{uo}0$ implies $\| Tx_{n}\|\rightarrow 0$, for each sequence $(x_{n})_n\subseteq E$. For a net $(x_{\alpha})_{\alpha}\subseteq E$, if $x_{\alpha}\xrightarrow{un}0$ implies $Tx_{\alpha}\xrightarrow{un}0$, then $T$ is called an unbounded norm continuous operator. In this manuscript, we study some properties of these classes of operators and their relationships with the other classes of operators.