We study deformations of zero-dimensional complete intersections in the plane, and prove the following results. (1) Two complex non-singular curves intersecting at r points with multiplicities d1, …, dr can be deformed into curves intersecting (at some points) with multiplicities d′1, …, d′s which are arbitrary prescribed partitions of the numbers d1, …, dr. (2) Two real curves intersecting with multiplicity at most 2 at each of their real common points can be deformed so that all real multiple intersection points split into real simple intersection points.