In this talk, I will introduce some flexible new bivariate copulas, the R package CopulaOne, and a few applications for data analytics in insurance and finance. Popular multivariate copulas such as vine copulas and factor copulas are constructed based on bivariate copulas. An ideal bivariate copula should have the following features. First, both upper and lower tails are able to explain full-range tail dependence. That is, the dependence in each tail can range among quadrant tail independence, intermediate tail dependence, and usual tail dependence. Second, it can capture upper and lower tail dependence patterns that are either the same or different. In this talk, I will discuss a general approach for constructing copulas that have the above features. Some promising parametric copula families are to be presented, and both the ideal features and the computational speeds were considered when constructing the copulas. Finally, a few applications using the copulas are to be demonstrated.