A geometrical approach to nonlinear perturbations in relativistic cosmology

16 pages; invited review article for Classical and Quantum Gravity issue on non-linear cosmologyInternational audienceWe give a pedagogical review of a covariant and fully non-perturbative approach to study nonlinear perturbations in cosmology. In the first part, devoted to cosmological fluids, we define a nonlinear extension of the uniform-density curvature perturbation and derive its evolution equation. In the second part, we focus our attention on multiple scalar fields and present a nonlinear description in terms of adiabatic and entropy perturbations. In both cases, we show how the formalism presented here enables one to easily obtain equations up to second, third and higher orders