Euclidean lattices are mathematical objects of increasing interest in the fields of cryptography and error-correcting codes. This doctoral thesis is a study on high-dimensional lattices with the motivation to understand how efficient they are in terms of being able to pack spheres. We study this by establishing a formula for the average number of lattice points of random Euclidean lattices inside a measurable subset of a real vector space, given the constraint that all the lattices are invariant under a prescribed finite group of symmetries. The thesis includes the discussion on what could be the appropriate probability space of random lattices with prescribed symmetries, when it is possible to derive an integration formula on these spaces and finally, and what the integration formula is given these conditions. The thesis then proceeds with an outline of recent applications of these integration formulas for the lattice packing problem. The techniques used involve number theory, representation theory, geometry and dynamics which the reader is introduced to in the text.