This talk will be a survey of the Levin-Gacs theory of
uniform randomness tests. In a nutshell, uniform tests generalize
Martin-Löf’s approach to randomness, by allowing tests not just for a
particular measure, but for a whole class of measures at once. We will
show how to construct such tests and present the main results of this
theory, such as randomness preservation under deterministic
transitions (and randomness of pre-images of random objects after such
transitions), and the existence of a neutral measure for compact sets.
If times permits, we will also discuss some applications to current
areas of interest, like random closed sets, or subsets of random sets.