Despite its long and illustrious story, the concept of symmetry is still evolving. A modern perspective on symmetry identifies the topological invariance of symmetry operators within correlation functions as their defining property. In addition to suggesting generalised notions of symmetry, this enables a calculus of topological defects that draws from well-established methods from topological quantum field theory. In this talk, I will discuss this picture in the case of one-dimensional quantum lattice models. Specifically, I will present a systematic way of constructing symmetric Hamiltonians as well as the corresponding (possibly non-invertible) lattice symmetry operators. I will then discuss a generalised gauging procedure of the symmetry, recovering celebrated duality relations, and the mapping of topological sectors upon such a gauging. Finally, I will comment about classification of gapped symmetric phases and the construction of the corresponding order parameters.