Universality is an abstract notion which relates to various mathematical contexts. Generally speaking, an object is called universal if, via a countable process, it can approximate every object in some class. In this talk we will present several examples which show that universal objects not only exist, but are generic rather than exceptional. Moreover we will talk about one of the most widely-studied instances of universality: the case of universal series, where the approximation is obtained by considering subsequences of partial sums. In particular, we will focus on holomorphic functions, and harmonic functions of several variables, which possess universal polynomial expansions.