Abstract Members of the family of pulsar-like stars are distinguished by their different manifestations observed, i.e., radio pulsars, accretion-driven X-ray pulsars, X-ray bursts, anomalous X-ray pulsars/soft gamma-ray repeaters, certral compact objects, and dim thermal neutron stars. Though one may conventionally think that these stars are normal neutron stars, it is still an open issue whether they are actually neutron stars or quark stars, as no convincing work, either theoretical from first principles or observational, has confirmed Baade-Zwicky's original idea that supernovae produce neutron stars. After introducing briefly the history of pulsars and quark stars, the author summarizes the recent achievements in his pulsar group, including quark matter phenomenology at low temperature, starquakes of solid pulsars, low-mass quark stars, and the pulsar magnetospheric activities.
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