We study the asymptotics of the number of standard Young tableaux of large skew diagrams. We present new bounds and discuss how they compare with the existing general bounds. Our approach is based on a recent Naruse's hook-length formula which I will also explain, and is connected to weighted lozenge tilings and a new particle model of independent interest. The talk assumes no prior knowledge of the subject. Based in part on joint work with Alejandro Morales, Greta Panova, Martin Tassy and Damir Yeliussizov.