Maximum G Edge-Packing (EPackG) is the problem of finding the maximum number of edge-disjoint isomorphic copies of a fixed guest graph G in a host graph H. This paper investigates the computational complexity of edge-packing for planar guests and planar hosts. Edge-packing is solvable in linear time when G is a 2-path and H is arbitrary, or when H is outerplanar and G is either a 3-cycle or a k-star. Edge-packing is solvable in polynomial time when both G and H are trees. Edge-packing is NP-complete when H is planar and G is either a cycle or a tree with ≥3 edges. The approximability of EPackG is considered. A strategy for developing polynomial-time approximation algorithms for planar hosts is exemplified by a linear-time approximation algorithm for EPack k-star that finds an edge-packing of size at least one-half optimal. Finally, EPackG is shown not to be in the complexity class Max SNP, though it is Max SNP-hard for G a k-star, ≥3.