In this talk we will focus on a ﬁeld of research within mathematical ﬁnance which known as market microstructure. This area of research deals with issues of market structure and design, price formation, transaction costs and investor behavior, among others. Modern ﬁnancial markets involve a range of participants who place buy and sell orders across a wide spectrum of time scales. We have pension funds that rebalance their portfolio on an annual basis, and on the other side of the scale, automated market-making algorithms and high frequency trading ﬁrms that submit several thousands of orders per second. In our research we use mathematical tools, mostly from probability theory and stochastic analysis, in order to model the behavior of these diﬀerent types of market participants, who are interacting with each other. From the analysis of the models, we often deduce explanations and quantitative description of various macro phenomena in the stock market. For example we design models in order to explain market volatility, price dynamics, transaction costs and to investigate the reasoning behind the repeated occurrence of ’ﬂash crashes’.