Normal neutrosophic numbers (NNNs) are a significant tool of describing the incompleteness, indeterminacy and inconsistency of the decision-making information. In this paper, we firstly propose the definition and the properties of the NNNs, and the accuracy function, the score function and the operational laws of the NNNs are developed. Then, some operators are presented, including the normal neutrosophic Bonferroni mean (NNBM) operator, the normal neutrosophic weighted Bonferroni mean (NNWBM) operator, the normal neutrosophic geometric Bonferroni mean (NNGBM) operator, the normal neutrosophic weighted geometric Bonferroni mean (NNWGBM) operator. We also study their properties and special cases. Further, we put forward a multiple attribute decision making method which is based on the NNWBM and NNWGBM operators. Finally, an illustrative example is given to verify the practicality and validity of the proposed method.