Following the celebrated work of R. Arens \\\\\\\\\\\\\\\\begin{center} {\\\\\\\\\\\\\\\\small ``{\\\\\\\\\\\\\\\\sc The adjoint of a bilinear operation}, {\\\\\\\\\\\\\\\\em Proc. Amer. Math. Soc.} {\\\\\\\\\\\\\\\\bf 2} (1951), 839-848\\\\\\\\\\\\\\\"}, \\\\\\\\\\\\\\\\end{center} the second dual $A^{**}$ of a Banach algebra $A$ enjoys two natural but, in general, different products (were termed the Arens products) each turning $A^{**}$ into a Banach algebra. During the last several years there has been a noticeable increase of interest in the algebraic and topological properties of these products on the second dual of certain Banach algebras.\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ In this talk we intend to give a review of the recent researches on the topics related to: $\\\\\\\\\\\\\\\\bullet$ The equality of Arens products. $\\\\\\\\\\\\\\\\bullet$ The size of topological centers. $\\\\\\\\\\\\\\\\bullet$ The amenability and weak amenability of $A^{**}$. \\\\\\\\\\\\\\\\noindent And then we present some research problems for further researches.

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