We define searches for Heavy Neutral Leptons (HNLs) as searches for Dirac or Majorana fermions with sterile neutrino quantum numbers, that are heavy enough to not disrupt the simplest Big Bang Nucleosynthesis bounds and/or unstable on cosmological timescales: Typically HNLs have mass $\sim{}$ MeV or higher. Searches for these particles generically set bounds on the mixing between the HNL and the active neutrinos, as parametrized by the extended 3${\times }$4 PMNS matrix elements ${{\mathit U}}_{ {{\mathit \ell}} {{\mathit x}} }$ (see the "Neutrino mass, mixing and oscillations" review) where ${{\mathit \ell}}$ = ${{\mathit e}}$, ${{\mathit \mu}}$ or ${{\mathit \tau}}$, and we denote the HNL as ${{\mathit \nu}_{{x}}}$. While many measurements may be interpreted to place bounds on various combinations of these matrix elements, we quote below limits only for those cases in which one matrix element is assumed to be much larger than the other two, i.e. $\vert {{\mathit U}}_{ {{\mathit \ell}} {{\mathit x}} }\vert $ ${}\gg$ $\vert {{\mathit U}}_{ {{\mathit \ell}^{\,'}} {{\mathit x}} }\vert $ for ${{\mathit \ell}^{\,'}}{}\not=$ ${{\mathit \ell}}$ . Experimental searches make use of various different strategies, including e.g. resonance searches in missing mass decay distributions or specific final states, searches for lepton number violating decays, and trilepton signatures. The resulting bounds on ${{\mathit U}}_{ {{\mathit \ell}} {{\mathit x}} }$ are typically dependent on the HNL mass. The quoted limits below are either the best limit near an experimental kinematic threshold, or a characteristic value in the mass range of the experimental sensitivity.