More than a decade of dedicated experimental work on the collisional physics of protoplanetary dust has brought us to a point at which the growth of dust aggregates can - for the first time - be self-consistently and reliably modeled. In this article, the emergent collision model for protoplanetery dust aggregates, as well as the numerical model for the evolution of dust aggregates in protoplanetary disks, is reviewed. It tums out that, after a brief period of rapid collisional growth of fluffy dust aggregates to sizes of a few centimeters, the protoplanetary dust particles are subject to bouncing collisions, in which their porosity is considerably decreased. The model results also show that low-velocity fragmentation can reduce the final mass of the dust aggregates but that it does not trigger a new growth mode as discussed previously.According to the current stage of our model, the direct formation of kilometer-sized planetesimals by collisional sticking seems unlikely, implying that collective effects,such as the streaming instability and the gravitational instability in dust-enhanced re-gions of the protoplanetary disk, are the best candidates for the processes leading to planetesimals.