Our goal is to explain how certain basic representation theoretic ideas and constructions encapsulated in the form of Macdonald processes lead to nontrivial asymptotic results in various `integrable'; probabilistic problems. Examples include dimer models, general beta random matrix ensembles, and various members of the (2+1)d anisotropic KPZ and (1+1)d KPZ universality classes, such as growing stepped surfaces, q-TASEP, q-PushASEP, and directed polymers in random media. No prior knowledge of the subject will be assumed.