This paper is based on the readings in the author's independent study on "advanced dynamical systems", and the author's mathematics honors project. It is a combination of the survey of some classical papers and the results from the research project. In the review part, none of the results are new and even less of them are due to the author; in the research part, we mainly focus the dynamics of the quadratic family along the real line. More specifically, in this paper we review and summarize the dynamics of one- and two- dimensional real quadratic maps from both topological and statistical viewpoints, and provide global pictures for their dynamics. Meanwhile, we briefly review the main results of the dynamics of one-dimensional complex quadratic maps under holomorphic singular perturbations, and provide recent research results about its dynamics under a nonholomorphic singular perturbation.