They have been flown on the FOTON-M3 mission...
In 2007, three projects were conducted during the FOTON-M3 mission
studies. The Tardigrade Resistance to Space Effects (TARSE) Project
was the first one involved in the mission of FOTON-M3. Its aim was to
analyse the impact of environmental stress, life history traits and
DNA damages in space (on board the spacecraft) on eutardigrade
Paramacrobiotus richtersi (Murray, 1911). In this project active and
anhydrobiotic tardigrades were exposed to radiation in microgravity
conditions. Both active and inactive individuals had high survival
rates with no induction of HSPs while showing an induction of the
antioxidant response (Rebecchi et al. 2009c, 2010b, 2011a). The next
project involved in the mission of FOTON-M3 was TARDIS (Tardigrada In
Space). The main goal of this project was to check whether tardigrades
from two species, Milnesium tardigradum Doyère, 1840 and Richtersius
coronifer (Richters, 1903), were able to survive conditions of open
space. The experiments showed that tardigrades can survive exposure to
the space vacuum, but the addition of factors such as ultraviolet
solar radiation, ionising solar radiation and galactic cosmic
radiation significantly reduced their survival rate (Jönsson, et al.
2008). In the third project from the FOTON-M3 mission, RoTaRad
(Rotifers, Tardigrades and Radiation), scientists examined effects on
initial survival, long-term survival and fecundity of selected species
of limno-terrestrial tardigrades in extreme stress conditions (mainly
cosmic radiation) (Persson et al. 2011).
and on the shuttle/ISS....
Next was the Endeavour mission in 2011 and the project TARDIKISS
(Tardigrades in Space). The main aim of this project was to broaden
our knowledge of life history traits and mechanisms of repairing
structural DNA damage during exposure to space flight stresses
(Rebecchi et al. 2011b; Vukich et al. 2012). The first results showed
that microgravity and cosmic radiation did not significantly affect
the survival rate of tardigrades (Rebecchi et al. 2011b; Vukich et al.
2012). However, Rizzo et al. (2015) showed a significant difference in
activities of ROS scavenging enzymes, the total content of glutathione
and the fatty acid composition between tardigrades sent into space and
control animals on Earth.
Additional info on the shuttle/ISS TARDIKISS experiments here.