  A net $(x_\alpha)$ in a vector lattice $X$ is unbounded order convergent to
$x \in X$ if $\lvert x_\alpha - x\rvert \wedge u$ converges to $0$ in order for
all $u\in X_+$. This convergence has been investigated and applied in several
recent papers by Gao et al. It may be viewed as a generalization of almost
everywhere convergence to general vector lattices. In this paper, we study a
variation of this convergence for Banach lattices. A net $(x_\alpha)$ in a
Banach lattice $X$ is unbounded norm convergent to $x$ if $\lVert\lvert
x_\alpha - x\rvert \wedge u\rVert\to 0$ for all $u\in X_+$. We show that this
convergence may be viewed as a generalization of convergence in measure. We
also investigate its relationship with other convergences.
