We study inflationary models where the kinetic sector of the theory has a non linearly realised symmetry which is broken by the inflationary potential. We distinguish between kinetic symmetries which non-linearly realise an internal or space-time group, and which yield a flat or curved scalar manifold. This classification leads to well-known inflationary models such as monomial inflation and alpha-attractors, as well as a new model based on fixed couplings between a dilaton and many axions which non-linearly realises higher-dimensional conformal symmetries. In this model, inflation can be realised along the dilatonic direction, leading to a tensor-to-scalar ratio r similar to 0.01 and a spectral index n(s) similar to 0.975. We refer to the new model as ambient inflation since inflation proceeds along an isometry of an anti-de Sitter ambient space-time, which fully determines the kinetic sector.