This paper presents a survey on some recent advances in the error rate control in multiple testing methodology. We consider the problem of controlling the $k$-family-wise error rate (kFWER, probability to make $k$ false discoveries or more) and the false discovery proportion (FDP, proportion of false discoveries among the discoveries), the latter being controlled either via its expectation, which is the so-called false discovery rate (FDR), or via its upper-tail distribution function. The emphasis is put both on the results which we want as unified as possible and on the mathematical proofs which we want as concise and simple as possible. This way, this paper is intended to a possibly non-specialist reader which would like to recover state-of-the-art multiple testing results using short proofs. As an additional contribution, we extend some previous work for controlling the kFWER and the upper-tail distribution of the FDP.