From their formation onto tiny solid particles, droplets must grow of more than three orders of magnitude in size, in order to become raindrops. To bridge this gap, two main physical mechanisms are at work in warm clouds. The first one is the growth by condensation of the surrounding vapor, dominant immediately after droplet formation. Later, when droplets become sufficiently large, collisions become relevant and droplets undergo a rapid growth yielding rain. The great efficiency of the collision/coalescence processes depends on the preceding condensation stage. In particular, since the probability of collision is proportional to the velocity difference between the two colliding droplets, it turns out that a population of identical droplets would be eventually unable to produce rain. A population of very similar droplets is exactly what one expects on the basis of a mean-field type analysis of the condensation equation. The narrowing of the droplet size distribution and the consequent bottleneck in droplet growth, are long-standing problems of cloud physics. A better understanding of these issues is needed in order to get reliable estimates of cloud lifetime, radiative balances and rain formation – important parameters of global climate models. I will focus on this problem and discuss numerical results suggesting that a broadening of droplet size distributions can be ascribed to turbulence.