Abstract: Following the Nobel prize in 2016, topological ideas are progressively enriching various research directions in condensed matter physics, from graphene to photonics. The recent years witnessed an exploding number of top-level publications with practical applications and fundamental physics. Topological photonics is attracting growing interest from classical and quantum perspectives. Authors reported many ground-breaking results, as topologically protected modes in photonic crystals and quantum simulations of the 4D Hall effect. Here we introduce the leading ideas in the most accessible way, and review topics like the topological classification of photonic band structures, synthetic magnetic fields, and artificial dimensions. We discuss some recent results on topological lasers and possible developments like frequency-comb emitters, PT-symmetric devices, and the application of machine learning to design topological photonic devices.