The intent of this paper is to introduce the concept of G(b)-cone metric and we describe some basic properties of such metric. Further, we establish some fixed point theorems for self-mappings satisfying the contractive type conditions and a common fixed theorem for two weakly compatible self-mappings satisfying the contractive condition in G(b)-cone metric spaces without the assumption of normality. Moreover, some examples are provided to illustrate the usability of the obtained results.