  We analyse TESS light curves for 70 southern $\lambda$ Boo stars to identify
binaries and to determine which of them pulsate as $\delta$ Scuti stars. We
find two heartbeat stars and two eclipsing binaries among the sample. We
calculate that 81 percent of $\lambda$ Boo stars pulsate as $\delta$ Sct
variables, which is about twice that of normal stars over the same parameter
space. We determine the temperatures and luminosities of the $\lambda$ Boo
stars from photometry and Gaia DR2 parallaxes. A subset of 40 $\lambda$ Boo
stars have 2-min TESS data, reliable temperatures and luminosities, and
$\delta$ Sct pulsation. We use Petersen diagrams (period ratios), \'echelle
diagrams and the period--luminosity relation to identify the fundamental mode
in 20 of those 40 stars and conclude that a further 8 stars are not pulsating
in this mode. For the remaining 12, the fundamental mode cannot be
unambiguously identified. Further mode identification is possible for 12 of the
fundamental mode pulsators that have regular sequences of pulsation overtones
in their \'echelle diagrams. We use stellar evolution models to determine
statistically that the $\lambda$ Boo stars are only superficially metal weak.
Simple pulsation models also better fit the observations at a metallicity of
$Z=0.01$ than at $Z=0.001$. The TESS observations reveal the great potential of
asteroseismology on $\lambda$ Boo stars, for determining precise stellar ages
and shedding light on the origin(s) of the $\lambda$ Boo phenomenon.
