For a Lie group
$G$
, we show that the map
$C_{c}^{\infty }\,\left( G \right)\,\times \,C_{c}^{\infty }\,\left( G \right)\,\to \,C_{c}^{\infty }\,\left( G \right),\,\left( \gamma ,\,\eta \right)\mapsto \,\gamma \,*\,\eta $
taking a pair of test functions to their convolution, is continuous if and only if
$G$
is
$\sigma -$
compact. More generally, consider
$r,\,s,\,t\,\in {{\mathbb{N}}_{0}}\,\cup \,\left\{ \infty \right\}$
with
$t\,\le \,r\,+\,s$
, locally convex spaces
${{E}_{1}}\,,\,{{E}_{2}}$
and a continuous bilinear map
$b:\,{{E}_{1}}\,\times \,{{E}_{2}}\,\to \,F$
to a complete locally convex space
$F$
. Let
$\beta :\,C_{c}^{r}\,\left( G,\,{{E}_{1}} \right)\,\times \,C_{c}^{S}\,\left( G,\,{{E}_{2}} \right)\,\to$
$C_{c}^{t}\,\left( G,\,F \right),\,\left( \gamma ,\,\eta \right)\,\mapsto \,\gamma \,*\,b\,\eta$
be the associated convolution map. The main result is a characterization of those
$\left( G,\,r,s,t,b \right)$
for which
$\beta$
is continuous. Convolution of compactly supported continuous functions on a locally compact group is also discussed as well as convolution of compactly supported
${{L}^{1}}$
-functions and convolution of compactly supported Radon measures.