There is as yet no firm evidence for quark matter in neutron stars. This is mainly because of the lack of direct probes of the opaque neutron star interior, and the lack of clear qualitative difference between hadronic and quark phases. The detection of GW170817 has offered a first example of how gravitational waves can be used to constrain the equation of state (EoS) of ultra-dense matter.
We shall discuss taking into account currently available information how to reveal possible phase transitions in neutron stars: the steadily growing body of astrophysical data and supported laboratory experiments should eventually allow us to narrow down the options by combining these various observations. We survey the proposed signatures of exotic matter, and emphasize the importance of data from neutron star mergers.