Since their introduction in a seminal paper by Hoeffding in 1948, U-statistics have played a major role in statistical science. Indeed, many important classical estimators and test statistics are of this form. Moreover, in more recent years, U-statistics have further been in frequent use in other fields like random geometry, big data and telecommunication networks, to name just a few. While in the classical situation, the kernel of the U-statistic remains fixed, modern applications often require that it may depend on the sample size n of the observed data. In this talk I will present several new results about the asymptotic normality of such U-statistics. The conditions for our results are phrased in terms of norms of so-called contraction kernels and are, hence, of an analytical nature. Moreover, since the arguments leading to our (functional) central limit theorems are based on Stein's method, they are also suitable for providing bounds on the rate of convergence. I will highlight the applicability of our results by discussing an example of subgraph counts in a random geometric graph model, which is also of relevance for goodness of fit testing. If time allows, then I will further discuss extensions to weighted and incomplete U-statistics as well as to general symmetric functionals of i.i.d. random variables. This talk is partly about joint work with Mikolaj Kasprzak and Giovanni Peccati from the University of Luxembourg.