Let $A$ be a Banach algebra, with second dual space $A""$. We propose to study the space $A""$ as a Banach algebra. There are two Banach algebra products on $A""$, denoted by $,Box,$ and $,Diamond,$. The Banach algebra $A$ is Arens regular if the two products $Box$ and $Diamond$ coincide on $A""$. In fact, $A""$ has two topological centers denoted by $mathfrak{Z}^{(1)}_t(A"")$ and $mathfrak{Z}^{(2)}_t(A"")$ with $A subset mathfrak{Z}^{(j)}_t(A"")subset A"";,(j=1,2)$, and $A$ is Arens regular if and only if $mathfrak{Z}^{(1)}_t(A"")=mathfrak{Z}^{(2)}_t(A"")=A""$. At the other extreme, $A$ is strongly Arens irregular if $mathfrak{Z}^{(1)}_t(A"")=mathfrak{Z}^{(2)}_t(A"")=A$. We shall give many examples to show that these two topological centers can be different, and can lie strictly between $A$ and $A""$.We shall discuss the algebraic structure of the Banach algebra $(A"",,Box,)V$; in particular, we shall seek to determine its radical and when this algebra has a strong Wedderburn decomposition. We are also particularly concerned to discuss the algebraic relationship between the two algebras $(A"",,Box,)$ and $(A"",,Diamond,)$. Most of our theory and examples will be based on a study of the weighted Beurling algebras $L^1(G,omega)$, where $omega$ is a weight function on the locally compact group $G$. The case where $G$ is discrete and the algebra is ${ell}^{,1}(G, omega)$ is particularly important.We shall also discuss a large variety of other examples. These include a weight $omega$ on $mathbb{Z}$ such that $ell^{,1}(mathbb{Z},omega)$ is neither Arens regular nor strongly Arens irregular, and such that the radical of $(ell^{,1}(mathbb{Z},omega)"", ,Box,)$ is a nilpotent ideal of index exactly $3$, and a weight $omega$ on $mathbb{F}_2$ such that two topological centers of the second dual of $ell^{,1}(mathbb{F}_2, omega)$ may be different, and that the radicals of the two second duals may have different indices of nilpotence.