  We introduce Lorentz spaces $L_{p(\cdot),q}(\R^n)$ and
$L_{p(\cdot),q(\cdot)}(\R^n)$ with variable exponents. We prove several basic
properties of these spaces including embeddings and the identity
$L_{p(\cdot),p(\cdot)}(\R^n)=L_{p(\cdot)}(\R^n)$. We also show that these
spaces arise through real interpolation between $L_{\p}(\R^n)$ and
$L_\infty(\R^n)$. Furthermore, we answer in a negative way the question posed
in Diening, H\"ast\"o, and Nekvinda (2004) whether the Marcinkiewicz
interpolation theorem holds in the frame of Lebesgue spaces with variable
integrability.
