We consider in this note one-side Liouville properties for viscosity solutions of various fully nonlinear uniformly elliptic inequalities, whose prototype is
F
(
x
,
D
2
u
)
≥
H
i
(
x
,
u
,
D
u
)
F(x,D^2u)\geq H_i(x,u,Du)
in
R
N
\mathbb {R}^N
, where
H
i
H_i
has superlinear growth in the gradient variable. After a brief survey on the existing literature, we discuss the validity or the failure of the Liouville property in the model cases
H
1
(
u
,
D
u
)
=
u
q
+
|
D
u
|
γ
H_1(u,Du)=u^q+|Du|^\gamma
,
H
2
(
u
,
D
u
)
=
u
q
|
D
u
|
γ
H_2(u,Du)=u^q|Du|^\gamma
and
H
3
(
x
,
u
,
D
u
)
=
±
u
q
|
D
u
|
γ
−
b
(
x
)
⋅
D
u
H_3(x,u,Du)=\pm u^q|Du|^\gamma -b(x)\cdot Du
, where
q
≥
0
q\geq 0
,
γ
>
1
\gamma >1
and
b
b
is a suitable velocity field. Several counterexamples and open problems are thoroughly discussed.