In recent decades urban scholarship has witnessed a ‘vertical’ or ‘volumetric’ turn that has advanced understandings of the multi-modal power asymmetries cutting through and organising urban space. Yet, this volumetric scholarship often remains locked into binary critiques – of success/failure, inclusion/exclusion, luxury/abjection, dispossession/accumulation, arborescent/rhizomatic, horizontal/vertical. This special issue tinkers with the limitations of these (unwittingly) binary urban geometries and volumetrologies – material as well as metaphorical ones. By building the etymological opposition of ‘the vertical’ with ‘the radical’ into the title of the volume (via the Latin root radix, meaning ‘root’), we seek to make the radical itself work with geometric and morphological associations. The papers in this special issue proffer diverse ethnographic, geographic and conceptual material for considering and theorising urban verticality in concert with rather than in opposition to its incumbent horizontalisms, diagonals, curls, zigzags and scattered planes. As we completed work on the special issue, the horrors of russia’s full-scale invasion of Ukrainian territory played out before our eyes. Accordingly, we make use of the introduction to reflect upon the insight that the war in Ukraine brings to bear on the intersection between domains of the urban, the vertical and the radical in the fraught, tense, vicious, fragile – but resistant – urban worlds of today. In doing so, we seek not only to render more clearly visible the violent effects of power verticals on lives, worlds and cities, but also to find seeds of hope in emergent, insurgent forms of (vertical as well as horizontal, and neither vertical nor horizontal) resistance.