Zagreb eccentricity indices were proposed analogously to Zagreb indices already known and used for almost forty years. For a connected graph, the first Zagreb eccentricity index is defined as the sum of the squares of the eccentricities of the vertices, and the second Zagreb eccentricity index is defined as the sum of the products of the eccentricities of pairs of adjacent vertices. We report mathematical properties, especially lower and upper bounds of trees and general graphs in terms of graph invariants and the corresponding extremal graphs, Nordhaus-Gaddum-type results, and the ordering of trees with small and large Zagreb eccentricity indices. (doi: 10.5562/cca1801)