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In this review presented at the Royal Society meeting, 'Cometary science after Rosetta', I present an overview of studies of small solar system objects that exhibit properties of both asteroids and comets (with a focus on so-called active asteroids). Sometimes referred to as 'transition objects', these bodies are perhaps more appropriately described as 'continuum objects', to reflect the notion that rather than necessarily representing actual transitional evolutionary states between asteroids and comets, they simply belong to the general population of small solar system bodies that happen to exhibit a continuous range of observational, physical and dynamical properties. Continuum objects are intriguing because they possess many of the properties that make classical comets interesting to study (e.g. relatively primitive compositions, ejection of surface and subsurface material into space where it can be more easily studied, and orbital properties that allow us to sample material from distant parts of the solar system that would otherwise be inaccessible), while allowing us to study regions of the solar system that are not sampled by classical comets.This article is part of the themed issue 'Cometary science after Rosetta'., (© 2017 The Author(s).)