The matter content of our universe is dominated by a component which behaves as a non-relativistic collisionless fluid, dubbed dark matter (DM). This statement assumes that General Relativity (GR) holds on cosmological scales, currently a reasonable hypothesis, while extensions of GR are still being investigated. Standard particle and nuclear physics tells us that DM cannot be made of known elementary particles. Yet, DM is a primary ingredient in our understanding of structure formation in the universe, as it seeds the gravitational wells that we observe today as galaxies. This makes the question of the origin of DM an exciting challenge in fundamental physics. After an introduction on the "cold DM paradigm", I will review the current research on particle candidates that arise in extensions of the standard model of elementary particles, originally motivated by issues inherent to particle physics. I will present the main classes of candidates (from exotic particles to exotic objects), the associated search strategies and current constraints, before discussing why and how the field has evolved toward more UV model-independent approaches, and related perspectives.