In this paper, we review two properties of completeness known as the Bourbaki-completeness and cofinal Bourbaki-completeness in the setting of metric spaces. These notions came from new classes of generalized Cauchy sequences appearing when characterizing the so-called Bourbaki-boundedness in a similar way that Cauchy sequences characterize the totally boundedness. For the clustering of Bourbaki–Cauchy sequences and cofinally Bourbaki–Cauchy sequences, we have respectively what is call Bourbaki-completeness and cofinal Bourbaki-completeness of metric spaces. The topological problem of metrizability by means of a Bourbaki-complete or a cofinally Bourbaki-complete metric has also been considered. Finally, we present detailed review of some relationships and mutual differences between these kinds of completeness.