We give a necessary and sufficient condition for equidistribution of continuous functions along monotone compact covers on locally compact spaces. We show the existence of equidistributed mappings along Bohr nets arising from group actions. Using almost periodic means, we give an analogue of Weyl’s equidis- tribution criterion for continuous functions with values in arbitrary topological groups. We prove van der Corput’s inequality on the lattice Nm for vectors in Hilbert spaces, and use this inequality to extend Hlawka’s equidistribution theorem to functions on the lattice Nm (m ≥ 1) with values in arbitrary topolog- ical groups.