Abstract.Institutions formalise the intuitive notion of logical system, including syntax, semantics, and the relation of satisfaction between them. Our exposition emphasises the natural way that institutions can support deduction on sentences, and inclusions of signatures, theories, etc.; it also introduces terminology to clearly distinguish several levels of generality of the institution concept. A surprising number of different notions of morphism have been suggested for forming categories with institutions as objects, and an amazing variety of names have been proposed for them. One goal of this paper is to suggest a terminology that is uniform and informative to replace the current chaotic nomenclature; another goal is to investigate the properties and interrelations of these notions in a systematic way. Following brief expositions of indexed categories, diagram categories, twisted relations and Kan extensions, we demonstrate and then exploit the duality between institution morphisms in the original sense of Goguen and Burstall, and the ‘plain maps’ of Meseguer, obtaining simple uniform proofs of completeness and cocompleteness for both resulting categories. Because of this duality, we prefer the name ‘comorphism’ over ‘plain map’; moreover, we argue that morphisms are more natural than comorphisms in many cases. We also consider ‘theoroidal’ morphisms and comorphisms, which generalise signatures to theories, based on a theoroidal institution construction, finding that the ‘maps’ of Meseguer are theoroidal comorphisms, while theoroidal morphisms are a new concept. We introduce ‘forward’ and ‘semi-natural’ morphisms, and develop some of their properties. Appendices discuss institutions for partial algebra, a variant of order sorted algebra, two versions of hidden algebra, and a generalisation of universal algebra; these illustrate various points in the main text. A final appendix makes explicit a greater generality for the institution concept, clarifies certain details and proves some results that lift institution theory to this level.