  We present several characterizations of uo-convergent nets or sequences in
spaces of continuous functions $C(\Omega)$, $C_b(\Omega)$, $C_0(\Omega)$, and
$C^\infty(\Omega)$, extending results of [vdW18]. In particular, it is shown
that a sequence uo-converges iff it converges pointwise on a co-meagre set. We
also characterize order bounded sets in spaces of continuous functions. This
leads to characterizations of order convergence.
