This work is concerned with an extension of the so-called Levi-Civita and Kustaanheimo-Stiefel transformations. The extension is achieved along two lines. Firstly, the authors examine the latter transformations in other dimensions than the ones originally considered by, on the one hand, Levi-Civita and, on the other hand, by Kustaanheimo and Stiefel. Secondly, they pass from the compact to the non-compact case. This leads to quadratic non-bijective transformations that they refer to as Hurwitz (or Kustaanheimo-Stiefel-like) and quasiHurwitz (or Levi-Civita-like) transformations. The Hurwitz and quasiHurwitz transformations are introduced and studied in an algebraic framework which relies on the use of (eight-dimensional) Cayley-Dickson algebras. An explicit formulation of the Hurwitz and quasiHurwitz transformations is also given in terms of Clifford algebras. The Hurwitz transformations are investigated from a geometrical viewpoint. Indeed, they are connected to Hopf and 'pseudoHopf' fibrations. Finally, some differential aspects of the Hurwitz and (to a lesser extent) quasiHurwitz transformations are developed in view of future physical applications.