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This paper first considers several types of additive bases. A typical problem is to find , the largest n for which there exists a set of distinct integers modulo n such that each r in the range can be written at least once as (modulo n) with . For example, , as illustrated by the set {0, 1, 2, 4, 8, 13, 18, 22}. The other problems arise if at least is changed to at most, or to , or if the words modulo n are omitted. Tables and bounds are given for each of these problems. Then a closely related graph labeling problem is studied. A connected graph with n edges is called harmonious if it is possible to label the vertices with distinct numbers (modulo n) in such a way that the edge sums are also distinct (modulo n). Some infinite families of graphs (odd cycles, ladders, wheels, ) are shown to be harmonious while others (even cycles, most complete or complete bipartite …