Crystal structures and physical properties of four families of Al-rich ternary uranium compounds with transition metals (TE) are reviewed, namely UTE2Al20, UTE2Al10, U6TE4Al43, and U3TE4Al12Al. The compounds can be described as consisting of 1 (isolated), 2 (dumbbells) or 3 (triangles) uranium atom clusters, surrounded (1-2-20, 1-2-10 and 6-4-43) or not (3-4-12) by large cages, which strongly influence their magnetic and related properties. Indeed, the ground states of the described systems evolve from Curie-like paramagnetism in the case of the phases with well-isolated, single U-atoms, to complex magnetic order or possible frustrated magnetism in the case of the systems with uranium triangles forming a breathing kagome lattice. We argue that the four families of uranium aluminides described in this review provide a unique opportunity to study magnetic interactions between U magnetic moments while gradually increasing the number of their nearest magnetic neighbors, and may also be helpful in understanding the fundamental origin of magnetic freezing phenomena.