Lie algebroids provide a rather flexible theory. They are relevant for describing many geometric structures such as Poisson, symplectic, or contact manifolds.

I will expose the notion of extension for Lie algebroids, and explain how they naturally get involved when studying fibrations with an extra structure on the fibers.

Ehresmann connections will be introduced in this context, for which the usual notions of curvature and parallel transport make sense. This approach allows a better understanding of the cohomology of an extension. It also provides a nice way to describe the topological groupoid integrating an extension. We will point out a lack of exactness of the “integration functor”, and explain the geometric counterpart.