We discuss the rainbow Ramsey theorems at limit cardinals and successors of singular cardinals, addressing some questions by Uri Abraham, James Cummings, and Clifford Smyth [J. Symbolic Logic 72 (2007), pp. 865–895] and Uri Abraham and James Cummings [Cent. Eur. J. Math. 10 (2012), pp. 1004–1016]. In particular, we show for inaccessible
κ
\kappa
,
κ
→
p
o
l
y
(
κ
)
2
−
b
d
d
2
\kappa \to ^{poly}(\kappa )^2_{2-bdd}
does not characterize weak compactness and for any singular cardinal
κ
\kappa
,
◻
κ
\square _\kappa
implies
κ
+
↛
p
o
l
y
(
η
)
>
κ
−
b
d
d
2
\kappa ^+\not \to ^{poly} (\eta )^2_{>\kappa -bdd}
for any
η
≥
c
f
(
κ
)
+
\eta \geq cf(\kappa )^+
and
κ
>
c
f
(
κ
)
=
κ
\kappa ^{>cf(\kappa )}=\kappa
implies
κ
+
→
p
o
l
y
(
ν
)
>
κ
−
b
d
d
2
\kappa ^+\to ^{poly} (\nu )^2_{>\kappa -bdd}
for any
ν
>
c
f
(
κ
)
+
\nu >cf(\kappa )^+
. We also provide a simplified construction of a model for
ω
2
↛
p
o
l
y
(
ω
1
)
2
−
b
d
d
2
\omega _2\not \to ^{poly} (\omega _1)^2_{2-bdd}
originally constructed by Uri Abraham and James Cummings [Cent. Eur. J. Math. 10 (2012), pp. 1004–1016] and show the witnessing coloring is indestructible under strongly proper forcings but destructible under some c.c.c forcing. Finally, we conclude with some remarks and questions on possible generalizations to rainbow partition relations for triples.