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In the paper, we recall the Wallman compactication 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 compactication of T. Along the way, we attempt to justify the advantages of the Wallman compactication over other manifestations of the Stone-Cech ompactication. The main result of the paper is a new form of the Arzela-Ascoli theorem, which introduces the concept of equicontinuity along ω-ultralters.Mathematics Subject Classication (2010): 54D35, 54D80.Key words: Arzela-Ascoli theorem, Wallman compactication, Stone-Cech compactica- tion, ultralters.