In this paper, our current studies on nonlinearity and blown-ups are summarized. Through comparing linear and nonlinear evolutions, it is observed that discontinuities and singularities are the fundamental characteristics of nonlinear whole evolutions. By combining mathematical models and the underlying physical systems, the concept of blown-ups is formally introduced for the study of whole evolutions. Based on analysis of nonlinear models, mathematical physics meanings and mathematical characters of blown-ups are provided.