This article studies the problem of distributed framework matching (FM), which originates from the assignment task in multirobot coordination and the matching task in pattern recognition. The objective of distributed FM is to distributively seek a correspondence which minimizes some metrics describing the disagreement between two frameworks (i.e., graphs and their embeddings). In view of the type of the underlying graph in the framework, two formulations, undirected framework matching (UFM) and directed framework matching (DFM), and their convex relaxations, relaxed UFM (RUFM), and relaxed DFM (RDFM), are presented. UFM is converted into a graph matching (GM) problem with the adjacency matrix being replaced by a matrix constructed from the undirected framework under certain graphical conditions, and can be solved distributively. Sufficient conditions for the equivalence between UFM and RUFM, and the perturbation admitting exact recovery of correspondence are established. On the other hand, DFM embeds the configuration of the directed framework via another type of matrix, whose computation is distributed, and can deal with the case of two frameworks with different sizes of node sets. A distributed optimization algorithm for solving RDFM is proposed and its convergence results are established which allows DFM to be solved in a fully distributed manner. Simulation examples on both synthetic data and real world datasets demonstrate the applicability and efficacy of our theoretical results in formation control and object matching problems.