This dissertation develops a theory of infinitesimal groupoids in algebraic geometry, a natural generalisation of algebraic foliations to the singular or positive characteristic case. Three applications are presented thereafter. The first one is a criterion to establish when the quotient of a scheme by a smooth connected equivalence relation is a scheme. The second one establishes finite generation of the invariant sections of a smooth groupoid in some special cases. The third one concerns a strategy to approximate formal solutions of differential equations by solutions on étale neighbourhoods.