Subword complexes are simplicial complexes introduced by A. Knutson and E. Miller as a tool to study Gröbner geometry of Schubert polynomials. In this talk, I will present some relevant results about of these objects in algebra, combinatorics, and geometry. In particular, I will focus on three applications: - a geometric construction of multi-associahedra, - a decisive result about denominator vectors in cluster algebras of finite type, and - a Hopf algebra structure on subword complexes. This talk is based on joint works with Nantel Bergeron, Jean-Philippe Labbé, Vincent Pilaud, and Christian Stump.