The key technical challenges for big data lie in how to discover the value of the low-value-density data and how to complete the task in the desired time. The ways to take up these challenges from three aspects were discussed. First is that the former challenge requires the combination of domain knowledge and data technology. This combination of theory and new algorithms forms the basis of application and analysis of big data. Second is that the latter challenge needs to design new types of computer, cluster system, computing framework, storage system and data management method, which forms the basis of computing and data of big data. Thirdly, both challenges relate to mathematical theory, which is the basis of mathematics of big data. In conclusion, several foundation issues for big data research including the basis of mathematics, computing, data, analysis and application of big data were analyzed.