In this paper we provide a concept of fuzzy partial metric space $(X,P,\ast )$ as an extension to fuzzy setting in the sense of Kramosil and Michalek, of the concept of partial metric due to Matthews. This extension has been defined using the residuum operator ${\to }_{\ast }$ associated to a continuous t-norm $\ast$ and without any extra condition on $\ast$. Similarly, it is defined the stronger concept of GV -fuzzy partial metric (fuzzy partial metric in the sense of George and Veeramani). After defining a concept of open ball in $(X,P,\ast )$, a topology ${\mathcal{T}}_P$ on X deduced from P is constructed, and it is showed that $(X,{\mathcal{T}}_P)$ is a $T_0$-space.