In this chapter, we discuss some of the main dynamical and statistical properties of the logistic map. We begin by analyzing the convergence to the fixed point at the bifurcations and in their neighborhoods. To do so, we use a set of scaling hypotheses, as well as a generalized homogeneous function that leads to a scaling law. Next, we discuss a route to chaos through the sequence of period-doubling bifurcations. We show that there is a ratio between the control parameters that identify the period-doubling bifurcations, which leads to one of the Feigenbaum exponents. We then introduce the concept of the Lyapunov exponent Lyapunov exponent and how it is calculated from the mapping equation for one-dimensional systems.