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General relativity has so far passed all experimental tests, with some of the most stringent tests in strong fields coming from observations of pulsars - rotating neutron stars that form from the collapsed cores of massive stars during supernovae. Such compact objects contain the densest form of matter in the observable universe, and therefore produce a strong gravitational field in their vicinity. The excellent rotational stability of pulsars makes them powerful tools for studying a wide range of topics in fundamental physics. This dissertation investigates four aspects using the new generation of radio telescopes: (1) constraining the dense matter equation of state by measuring the moment of inertia of neutron stars, (2) testing higher-order gravitational light propagation effects in the Double Pulsar system using observations from the MeerKAT telescope, (3) prospects of testing scalar-tensor gravity using pulsar-white dwarf system and hypothetical pulsar-black hole systems, and (4) recent advances in the detection of nanohertz gravitational waves using pulsar timing arrays and efforts to improve it.
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