Kuga varieties are a natural generalisation of universal families of abelian varieties. This thesisdescribes the candidate’s work on the geometry of some types of Kuga varieties. In Part I, byconsidering a special kind of Kuga varieties resulting from the Kuga-Satake construction, we constructan explicit map from a moduli space of K3 surfaces of Picard rank 14 to a moduli space of polarisedabelian 8-folds with totally definite quaternion multiplication. This is a geometric interpretation ofan exceptional coincidence between locally symmetric spaces of type II_4 and type IV_6. In Part II,we study the n-fold Kuga varieties associated to the moduli space of (1, p)-polarised abelian surfaceswith canonical level structure for prime p at least 3, and compute their Kodaira dimensions for allbut 27 possible combinations of (n, p).