Concrete is the most widely used construction material. Worldwide consumption of cement, the strength‐giving component in concrete, is now estimated to be 4.10 Gt per year, having risen from 2.22 Gt just 10 years ago. This rate of consumption means that cement manufacture alone is estimated to account for 5.2 % of global carbon dioxide emissions.Concrete offers the opportunity to create structures with almost any geometry economically. Yet its unique fluidity is seldom capitalized upon, with concrete instead being cast in rigid, flat moulds to create non‐optimized geometries that result in structures with a high material usage and large carbon footprints. This paper will explore flexible formwork construction technologies that embrace the fluidity of concrete to facilitate the practical construction of concrete structures with complex and efficient geometries.This paper presents the current state of the art in flexible formwork technology, highlighting practical uses, research challenges and new opportunities.