  With seven planets, the TRAPPIST-1 system has the largest number of
exoplanets discovered in a single system so far. The system is of
astrobiological interest, because three of its planets orbit in the habitable
zone of the ultracool M dwarf. Assuming the planets are composed of
non-compressible iron, rock, and H$_2$O, we determine possible interior
structures for each planet. To determine how much tidal heat may be dissipated
within each planet, we construct a tidal heat generation model using a single
uniform viscosity and rigidity for each planet based on the planet's
composition. With the exception of TRAPPIST-1c, all seven of the planets have
densities low enough to indicate the presence of significant H$_2$O in some
form. Planets b and c experience enough heating from planetary tides to
maintain magma oceans in their rock mantles; planet c may have eruptions of
silicate magma on its surface, which may be detectable with next-generation
instrumentation. Tidal heat fluxes on planets d, e, and f are lower, but are
still twenty times higher than Earth's mean heat flow. Planets d and e are the
most likely to be habitable. Planet d avoids the runaway greenhouse state if
its albedo is $\gtrsim$ 0.3. Determining the planet's masses within $\sim0.1$
to 0.5 Earth masses would confirm or rule out the presence of H$_2$O and/or
iron in each planet, and permit detailed models of heat production and
transport in each planet. Understanding the geodynamics of ice-rich planets f,
g, and h requires more sophisticated modeling that can self-consistently
balance heat production and transport in both rock and ice layers.
