Let be a separating family of lattice seminorms on a vector lattice X, then is called a multi-normed vector lattice (or MNVL). We write if for all . A net in an MNVL is said to be unbounded m-convergent (or um-convergent) to x if for all . um-Convergence generalizes un-convergence (Deng et al. in Positivity 21:963-974, 2017; KandiAc et al. in J Math Anal Appl 451:259-279, 2017) and uaw-convergence (Zabeti in Positivity, 2017. doi:10.1007/s11117-017-0524-7), and specializes up-convergence (AydA +/- n et al. in Unbounded p-convergence in lattice-normed vector lattices. arXiv:1609.05301) and -convergence (Dabboorasad et al. in -Convergence in locally solid vector lattices. arXiv:1706.02006v3). 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 , 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 -Lebesgue and -Levi properties in terms of sequential um-completeness. Finally, we prove that every m-bounded and um-closed set is um-compact iff the space is atomic and has Lebesgue's and Levi's properties.