We introduce the following optimization version of the classical pattern matching problem (referred to as the maximum pattern matching problem). Given a two-dimensional rectangular text and a two-dimensional rectangular pattern, find the maximum number of non-overlapping occurrences of the pattern in the text. Unlike the classical two-dimensional pattern matching problem, the maximum two-dimensional pattern matching problem is NP-complete. We devise polynomial time approximation algorithms and approximation schemes for this problem. We also briefly discuss how the approximation algorithms can be extended to include a number of other variants of the problem.