We consider an stochastic, individual-based model of an evolving population with logistic density-dependence, where individuals are characterized by a quantitative phenotypic trait. Under appropriate parameters scalings of rare mutations and large populations, we obtain a stochastic jump process on the mutation time-scale, where evolution proceeds through successive invasions of mutants, followed by competition phases on shorter time scales, where disadvantaged traits are eliminated. Under an additional scaling of small mutations and on an appropriate time scales, the evolution can be described as ordinary differential equations on the trait space, known as "canonical equations of adaptive dynamics", followed by diversification phases where the number of traits present in the population may increase, a phenomenon known as "evolutionary branching". This is joint work with Sylvie M�l�ard (Ecole Polytechnique).