Using molecular dynamics computer simulations we investigate how the glass transition and the properties of the resulting glass depend on the cooling rate with which the sample has been quenched. The system we study is a two component Lennard-Jones model which is coupled to a heat bath whose temperature is decreased from a high temperature, where the system is a liquid, to zero temperature, where the system is a glass. The temperature T-b Of this heat bath is decreased linearly in time, i.e. T-b = T-i - gamma t, where gamma is the cooling rate, and we study the cooling rate dependence by varying gamma over several orders of magnitude. In accordance with simple theoretical arguments and with experimental observations we find that the glass transition, as observed in the specific heat and the thermal expansion coefficient, becomes sharper when gamma is decreased. A decrease of the cooling rate also leads to a decrease of the glass transition temperature T-g and we show that the dependence of T-g on gamma can be rationalized by assuming that the temperature dependence of the relaxation times of the system is given by either a Vogel-Fulcher law or a power law. By investigating the structural properties of the glass, such as the radial distribution functions, the coordination numbers and the angles between three neighbor-sharing particles, we show how the local order of the glass increases with decreasing cooling rate. The enthalpy H and the density rho of the glass decrease and increase, respectively, with decreasing gamma. By investigating the gamma dependence of clusters of nearest neighbors, we show how the cooling rate dependence of H and rho can be understood from a microscopic point of view. Furthermore we demonstrate that the frequency of icosahedral-like structures is decreasing with decreasing cooling rate. We also show that the spectrum of the glass, as computed from the dynamical matrix, shows a shift towards higher frequencies when gamma is decreased. All these effects show that there is a significant dependence of the properties of glasses on the cooling rate with which the glass is produced.