In this paper we determine the
Q
\mathbb {Q}
-gonalities of the modular curves
X
0
(
N
)
X_0(N)
for all
N
>
145
N>145
. We determine the
C
\mathbb {C}
-gonality of many of these curves and the
Q
\mathbb {Q}
-gonalities and
C
\mathbb {C}
-gonalities for many larger values of
N
N
.
Using these results and some further work, we determine all the modular curves
X
0
(
N
)
X_0(N)
of gonality
4
4
,
5
5
and
6
6
over
Q
\mathbb {Q}
. We also find the first known instances of pentagonal curves
X
0
(
N
)
X_0(N)
over
C
\mathbb {C}
.