Summary: It is well known that if \(X\) and \(Y\) are completely regular \(T_2\) spaces, then any continuous function, \(f\), from \(X\) to \(Y\), has a unique continuous extension, \(\beta(f)\), from \(\beta X\) to \(\beta Y\), where \(\beta X\) and \(\beta Y\) are the Stone-Čech compactifications of \(X\) and \(Y\), respectively. This function plays an important role in Stone-Čech theory, especially in questions pertaining to embeddability. We first extend this construction to general Wallman spaces, and then apply the results to extend well-known embeddability theorems.