Let
C
C
be the cusp
{
(
x
,
y
)
:
x
≥
0
\{ (x,y):x \geq 0
,
−
x
β
≤
y
≤
x
β
}
- {x^\beta } \leq y \leq {x^\beta }\}
where
β
>
1
\beta > 1
. Set
∂
C
1
=
{
(
x
,
y
)
:
x
≥
0
,
y
=
−
x
β
}
\partial {C_1} = \{ (x,y):x \geq 0, y = - {x^\beta }\}
and
∂
C
2
=
{
(
x
,
y
)
:
x
≥
0
\partial {C_2} = \{ (x,y):x \geq 0
,
y
=
x
β
}
y = {x^\beta }\}
. We study the existence and uniqueness in law of reflecting Brownian motion in
C
C
. The angle of reflection at
∂
C
j
∖
{
0
}
\partial {C_j}\backslash \{ 0\}
(relative to the inward unit normal) is a constant
θ
j
∈
(
−
π
2
,
π
2
)
{\theta _j} \in \left ( { - \frac {\pi } {2},\frac {\pi } {2}} \right )
, and is positive iff the direction of reflection has a negative first component in all sufficiently small neighborhoods of
0
0
. When
θ
1
+
θ
2
≤
0
{\theta _1} + {\theta _2} \leq 0
, existence and uniqueness in law hold. When
θ
1
+
θ
2
>
0
{\theta _1} + {\theta _2} > 0
, existence fails. We also obtain results for a large class of asymmetric cusps. We make essential use of results of Warschawski on the differentiability at the boundary of conformal maps.