In this new era of powerful telescopes such as ALMA, we are now able to study the birth of planets in disks around young stars, in more detail than ever before. New observations are revealing fascinating structures in protoplanetary disks that are transforming our understanding of the formation and evolution of planetary systems. In this colloquium, I will explain theoretical models of dust evolution in protoplanetary disks and I will compare these theoretical predictions with current multi-wavelength disk observations. This link is providing significant insights about how different physical conditions play a crucial role in the formation of the first planetesimals, and is extending our understanding of how initial conditions of protoplanetary disks are reflected in the large diversity of extrasolar systems observed up today.