  A net $(x_\alpha)$ in a vector lattice $X$ is said to uo-converge to $x$ if
$|x_\alpha-x|\wedge u\xrightarrow{\rm o}0$ for every $u\ge 0$. In the first
part of this paper, we study some functional-analytic aspects of
uo-convergence. We prove that uo-convergence is stable under passing to and
from regular sublattices. This fact leads to numerous applications presented
throughout the paper. In particular, it allows us to improve several results in
[26,27]. In the second part, we use uo-convergence to study convergence of
Ces\`aro means in Banach lattices. In particular, we establish an intrinsic
version of Koml\'os' Theorem, which extends the main results of [35,16,31] in a
uniform way. We also develop a new and unified approach to Banach-Saks
properties and Banach-Saks operators based on uo-convergence. This approach
yields, in particular, short direct proofs of several results in [21,24,25].
