Elementary particles with continuous spin

Classical results and recent developments on the theoretical description of elementary particles with "continuous" spin are reviewed. At free level, these fields are described by unitary irreducible representations of the isometry group (either Poincar\'e or anti de Sitter group) with an infinite number of physical degrees of freedom per spacetime point. Their basic group-theoretical and field-theoretical descriptions are reviewed in some details. We mention a list of open issues which are crucial to address for assessing their physical status and potential relevance.