In this dissertation we describe connections between the equivariant theory of Lie groupoids and the non-commutative geometry of the convolution algebra. Lie groupoids are objects that encode symmetries of a space. They are a generalization of Lie groups in the following sense: Lie groups describe symmetries that are globally defined on the space, while Lie groupoids describe symmetries whose application is place-dependent. These objects are interesting because of their applications in physics. The existence of solutions to a physical system can be proven or disproven by exhibiting geometric properties of the space on which the system is applied. When there is a symmetry under which the system is invariant, you can shrink the space on which one needs to solve the system by factoring out the symmetry. In this philosophy we are interested in the ‘geometry of the space that is invariant under the symmetry’. In case where there is an action of a Lie group on the space, this is something we understand reasonably well, and in this dissertation we try the generalize these ideas to Lie groupoids. We want to understand the non-commutative geometry of convolution algebras, and that is the mathematical content of this dissertation. We describe connections between various mathematical properties of a Lie groupoid and the non-commutative geometry of the convolution algebra, with the goal to sketch a complete picture of this non-commutative geometry.