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.
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