The concept of Hartman-measurability was introduced in [1]. It turned out to be useful in the context of coding -sequences and related group-rotations. Among other things one can prove that there exists an unique translation invariant finite additive measure on Hartman-measurable subsets of the integers. We extend this concept from sets to functions via a process similar to the passage from measurable sets to measurable functions. The relation between this new class of Hartman-measurable functions and the class of almost periodic functions is comparable to the relation between Riemann integrable functions to continuous functions. We report on some results related to the extent of Hartman-measurable functions and establish a connection between Hartman-measurabilty and concepts such as (weak) almost periodicity.