The dense matter equation of state (EoS), describing the state of matter under the extreme conditions found in neutron stars, is not accurately known. However, significant process has been made in recent years through the emergence of new observational avenues of neutron stars. Firstly, the X-ray timing telescope NICER has delivered two joint mass-radius measurements, for pulsars PSR J0030+0451 and PSR J0740+6620, using pulse profile modeling. Secondly, gravitational wave detections of binary neutron star (BNS) mergers allow for a measurement of the EoS-dependent tidal deformability, as demonstrated in the first detected BNS merger GW170817. Additionally, electromagnetic radiation from the subsequent ultraviolet-optical-infrared transient (the kilonova) originating from the ejected material in GW170817 further probes the binary system and the EoS. We demonstrate how the joint analysis of these multi-messenger observations of neutron stars significantly constrains the dense matter EoS. We then describe, in more detail, a framework to jointly analyse a gravitational wave signal and the accompanying kilonova light curves, focusing on possible future black hole–neutron star (BHNS) mergers. We highlight the potential for multimessenger BHNS to constrain the tidal deformability of the neutron star, thereby increasing our understanding of the dense matter EoS.