  This paper presents a survey on some recent advances for the type I 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 FDP is controlled either via
its expectation, which is the so-called false discovery rate (FDR), or via its
upper-tail distribution function. We aim at deriving general and unified
results together with concise and simple mathematical proofs. Furthermore,
while this paper is mainly meant to be a survey paper, some new contributions
for controlling the kFWER and the upper-tail distribution function of the FDP
are provided. In particular, we derive a new procedure based on the quantiles
of the binomial distribution that controls the FDP under independence.
