The aim of this paper is to obtain general univalence conditions and quasiconformal extensions to ${\cal C}^n$ of holomorphic mappings defined on the Euclidian unit ball $B$. The asymptotical case of the quasiconformal extension results is also presented. We extend several results obtained by Hamada and Kohr (2011) in [15] to a more general case. In particular our results improve certain univalence criteria and quasiconformal extension results previously obtained by Pfaltzgraff [21], [22], Curt and Pascu [8], Curt [5], Hamada and Kohr [16], Curt and Kohr [6], [7] and Răducanu [24]. As applications we present general forms of the $n$-dimensional version of the well-known univalence criterion due to Lewandowski [19] and its quasiconformal extension.