Multiple dispatch – the selection of a function to be invoked based on the dynamic type of two or more arguments – is a solution to several classical problems in object-oriented programming. Open multi-methods generalize multiple dispatch towards open-class extensions, which improve separation of concerns and provisions for retroactive design. We present the rationale, design, implementation, and performance of a language feature, called open multimethods, for C++. Our open multi-methods support both repeated and virtual inheritance. Our call resolution rules generalize both virtual function dispatch and overload resolution semantics. After using all information from argument types, these rules can resolve further ambiguities by using covariant return types. Great care was taken to integrate open multi-methods with existing C++ language features and rules. We describe a model implementation and compare its performance and space requirements to existing open multi-method extensions...