  In this paper we establish quaternionic and octonionic analogs of the
classical Riemann surfaces. The construction of these manifolds has nice
peculiarities and the scrutiny of Bernhard Riemann approach to Riemann
surfaces, mainly based on conformality, leads to the definition of slice
conformal or slice isothermal parameterization of quaternionic or octonionic
Riemann manifolds. These new classes of manifolds include slice regular
quaternionic and octonionic curves, graphs of slice regular functions, the $4$
and $8$ dimensional spheres, the helicoidal and catenoidal $4$ and $8$
dimensional manifolds. Using appropriate Riemann manifolds, we also give a
unified definition of the quaternionic and octonionic logarithm and $n$-th root
function.
