Most distortion theorems for
K
K
-quasiconformal mappings in
R
n
{{\mathbf {R}}^n}
,
n
⩾
2
n \geqslant 2
, depend on both
n
n
and
K
K
in an essential way, with bounds that become infinite as
n
n
tends to
∞
\infty
. The present authors obtain dimension-free versions of four well-known distortion theorems for quasiconformal mappings—namely, bounds for the linear dilatation, the Schwarz lemma, the
Θ
\Theta
-distortion theorem, and the
η
\eta
-quasisymmetry property of these mappings. They show that the upper estimates they have obtained in each of these four main results remain bounded as
n
n
tends to
∞
\infty
with
K
K
fixed. The proofs are based on a "dimensioncancellation" property of the function
t
↦
τ
−
1
(
τ
(
t
)
/
K
)
,
t
>
0
,
K
>
0
t \mapsto {\tau ^{ - 1}}(\tau (t)/K),\,t > 0,\,K > 0
, where
τ
(
t
)
\tau (t)
is the capacity of a Teichmüller extremal ring in
R
n
{{\mathbf {R}}^n}
. The authors also prove a dimension-free distortion theorem for the absolute (cross) ratio under
K
K
-quasiconformal mappings of
R
¯
n
{\overline {\mathbf {R}} ^n}
, from which several other distortion theorems follow as special cases.