One of the classical topics in probability theory is the analysis of the fluctuation of partial sums of independent random variables. In the last decades many researchers tried to generalize the classical limit theorems (e.g. the law of large numbes, central limit theorem and the law of the iterated logarithm) to certain dependent structures. This is not only of theoretical interest but has also important applications, e.g. for statistical inference. In our research we develop and use strong invariance methods which allow us to derive several limit theorems under very sharp or even optimal conditions.