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A research monograph introducing circle systems. If G is an abelian group; g is an element of G and f is an injective mapping from G into the real projective plane such that for each four element subset {a,b,c,d} of G with sum g, the corresponding points {f(a),f(b),f(c),f(d)} are cocyclic, then the set of points f(G) and the associated cocyclic quadruples is a (G,g) circle system. The group G is the base of the system, and the element g is the sum. Circle systems with various bases and sums are constructed and their properties determined. In particular, it is shown that the points of a circle system lie on a self-inversive cubic or quartic algebraic curve which we call the envelope of the system. The ternary hypercommutative algebra which is defined on the envelope is used to study algebraic properties of circle systems. Circle systems with noncircular conic envelope are also studied.