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In the paper, we recall the Wallman compactification of a Tychonoff space T (denoted by Wall(T)) and the contribution made by Gillman and Jerison. Motivated by the Gelfand-Naimark theorem, we investigate the homeomorphism between Cb(T), the space of continuous and bounded functions on T, and C(Wall(T)), the space of continuous functions on the Wallman compactification of T. Along the way, we attempt to justify the advantages of the Wallman compactification over other manifestations of the Stone-Čech compactification. The main result of the paper is a new form of the Arzelà-Ascoli theorem, which introduces the concept of equicontinuity along ω-ultrafilters. [ABSTRACT FROM AUTHOR]