This paper describes the design, implementation, and evaluation of MultiJava, a backward-compatible extension to The Java Programming Language[Superscript TM] that supports open classes and symmetric multiple dispatch. An open class is one to which new methods can be added without editing the class directly. Multiple dispatch allows the method invoked by a message send to depend on the run-time types of any subset of the argument objects. MultiJava is the first full-scale programming language to support these features while retaining modular static typechecking and compilation. The paper defines the notions of modular editing, typechecking, and compilation, and describes two problems, the augmenting method problem and the binary method prob...