The notion of a multivariate local and tail empirical process indexed by sets is introduced which is notationally too involved to be repeated here. For such processes a number of functional laws of the iterated logarithm are established. The proofs are entirely based on the general theory of empirical processes and do not make use of strong invariance principles. The theorems obtained in this way lead to a unified approach to the study of the asymptotic almost sure behavior of various statistics which can be expressed as local functionals of the empirical distribution. Among these statistics are the multivariate Parzan- Rosenblatt kernel density estimator and the Bahadur-Kiefer representation.