  Let $\mathcal{M}=\{m_\lambda\}_{\lambda\in\Lambda}$ be a separating family of
lattice seminorms on a vector lattice $X$, then $(X,\mathcal{M})$ is called a
multi-normed vector lattice (or MNVL). We write $x_\alpha
\xrightarrow{\mathrm{m}} x$ if $m_\lambda(x_\alpha-x)\to 0$ for all
$\lambda\in\Lambda$. A net $x_\alpha$ in an MNVL $X=(X,\mathcal{M})$ is said to
be unbounded $m$-convergent (or $um$-convergent) to $x$ if $\lvert x_\alpha-x
\rvert\wedge u \xrightarrow{\mathrm{m}} 0$ for all $u\in X_+$. $um$-Convergence
generalizes $un$-convergence \cite{DOT,KMT} and $uaw$-convergence \cite{Zab},
and specializes $up$-convergence \cite{AEEM1} and $u\tau$-convergence
\cite{DEM2}. $um$-Convergence is always topological, whose corresponding
topology is called unbounded $m$-topology (or $um$-topology). We show that, for
an $m$-complete metrizable MNVL $(X,\mathcal{M})$, the $um$-topology is
metrizable iff $X$ has a countable topological orthogonal system. In terms of
$um$-completeness, we present a characterization of MNVLs possessing both
Lebesgue's and Levi's properties. Then, we characterize MNVLs possessing
simultaneously the $\sigma$-Lebesgue and $\sigma$-Levi properties in terms of
sequential $um$-completeness. Finally, we prove that any $m$-bounded and
$um$-closed set is $um$-compact iff the space is atomic and has Lebesgue's and
Levi's properties.
