In a 1988 paper, Y. Peres proved that in any probability-measure preserving system {X, mu, T}, where X is compact and T is continuous, coupled with a continuous function f, there is at least one point x (which we term heavy) such that 1ni=0 n-1f&j0;Ti&parl0;x&parr0;≥ Xfdm for all n ∈ N . We simplify and expand the proof to a more general result, and investigate the properties of the set of heavy points for a given T and f. The structure of the set of such points in irrational circle rotations is studied extensively, followed by a development of similar ideas in symbolic dynamics. Finally, the notion of heaviness is exported to arbitrary sequences, and a few results contrasting heaviness with equidistribution are developed. Open problems for future research are included throughout.