Aggregation & that is, the computation of global properties like average or maximal load, or the number of nodes & is an important basic functionality in fully distributed environments. In many cases & which include protocols responsible for self-organization in large-scale systems and collaborative environments & it is useful if all nodes know the value of some aggregates continuously. In this paper we present and analyze novel protocols capable of providing this service. The proposed anti-entropy aggregation protocols compute different aggregates of component properties like extremal values, average and counting. Our protocols are inspired by the anti-entropy epidemic protocol where random pairs of databases periodically resolve their differences. In the case of aggregation, resolving difference is generalized to an arbitrary (numeric) computation based on the states of the two communicating peers. The advantage of this approach is that it is proactive and "democratic", which means it has no performance bottlenecks, and the approximation of the aggregates is present continuously at all nodes. These properties make our protocol suitable for implementing e.g. collective decision making or automatic system maintenance based on global information in a fully distributed fashion. As our main contribution we provide fundamental theoretical results on the proposed averaging protocol.