  We present in this paper the construction of a continuum directed polymer
model in an environment given by space-time L\'evy noise. One of the main
objectives of this construction is to describe the scaling limit of discrete
directed polymer in an heavy-tail environment and for this reason we put
special emphasis on the case of $\alpha$-stable noises with $\alpha \in (1,2)$.
Our construction can be performed in arbitrary dimension, provided that the
L\'evy measure satisfies specific (and dimension dependent) conditions. We also
discuss a few basic properties of the continuum polymer and the relation
between this model and the Stochastic Heat Equation with multiplicative L\'evy
noise.
