  We introduce singular subalgebroids of an integrable Lie algebroid, extending
the notion of Lie subalgebroid by dropping the constant rank requirement. We
lay the bases of a Lie theory for singular subalgebroids: we construct the
associated holonomy groupoids, adapting the procedure of
Androulidakis-Skandalis for singular foliations, in a way that keeps track of
the choice of Lie groupoid integrating the ambient Lie algebroid. The holonomy
groupoids are topological groupoids, and are suitable for noncommutative
geometry as they allow for the construction of the associated convolution
algebras. Further we carry out the construction for morphisms in a functorial
way.
