  In this paper, we introduce an extension of rectangular metric spaces called
controlled rectangular metric spaces, by changing the rectangular inequality as
follows: \begin{equation*}
  d(x, y)\leq\alpha(x, u)d(x, u)+\alpha(u, v)d(u, v)+\alpha(v, y)d(v, y),
\end{equation*} for all distinct $x, y, u, v\in X$ with the function $\alpha:
X\times X\rightarrow[1,\infty[$. We also establish some fixed point theorems
for self-mappings defined on such spaces. Our main results extends and improves
many results existing in the literature. Moreover, an illustrative example is
presented to support the obtained results.
