Let
S
n
f
{S_n}f
be the
n
n
th partial sum of the Vilenkin-Fourier series of
f
∈
L
1
f \in {L^1}
. For
1
>
p
>
∞
1 > p > \infty
, we characterize all weight functions
w
w
such that if
f
∈
L
p
(
w
)
f \in {L^p}(w)
,
S
n
f
{S_n}f
converges to
f
f
in
L
p
(
w
)
{L^p}(w)
. We also determine all weight functions
w
w
such that
{
S
n
}
\{ {S_n}\}
is uniformly of weak type
(
1
,
1
)
(1,1)
with respect to
w
w
.