  Well-known results of Lagrange and Jacobi prove that the every $m \in \mathbb
N$ can be expressed as a sum of four integer squares, and the number $r(m)$ of
such representations can be given by an explicit formula in $m$. In this paper,
we prove that the only real quadratic number field for which the sum of four
squares is universal is $\mathbb Q(\sqrt{5})$. We provide explicit formulas for
$r(m)$ for $K= \mathbb Q (\sqrt{2})$ and $K= \mathbb Q(\sqrt{5})$. We then
consider the theta series of the sum of four squares over any real quadratic
number field, providing explicit upper and lower bounds for the Eisenstein
coefficients. Last, we include examples of the complete theta series
decomposition of the sum of four squares over $\mathbb Q (\sqrt{3})$, $\mathbb
Q (\sqrt{13})$ and $\mathbb Q (\sqrt{17})$.
