Summary: We present a new perspective on gravitational lensing. We describe a new extension of the weak lensing formalism capable of describing strongly lensed images. By integrating the nonlinear geodesic deviation equation, the amplification matrix of weak lensing is generalised to a sum over independent amplification tensors of increasing rank. We show how an image distorted by a generic lens may be constructed as a sum over ‘roulettes’, which are the natural curves associated with the independent spin modes of the amplification tensors. Highly distorted images can be constructed even for large sources observed near or within the Einstein radius of a lens where the shear and convergence are large. The amplitude of each roulette is formed from a sum over appropriate derivatives of the lensing potential. Consequently, measuring these individual roulettes for images around a lens gives a new way to reconstruct a strong lens mass distribution without requiring a lens model. This formalism generalises the convergence, shear and flexion of weak lensing to arbitrary order, and provides a unified bridge between the strong and weak lensing regimes. This overview paper is accompanied by a much more detailed paper II, [the author, “Roulettes: A weak lensing formalism for strong lensing. II: Derivation and analysis”, Preprint, arXiv:1603.04652].