We prove the existence and give estimates of the fundamental solution (the heat kernel) for the equation $\partial_t ={\mathcal L}^{\kappa}$ for non-symmetric non-local operators$${\mathcal L}^{\kappa}f(x):= \int_{\mathbb R^d}( f(x+z)-f(x)- {\bf 1}_{|z| < 1} \langle z,\nabla f(x) \rangle )\kappa(x,z)J(z)\, dz\,,$$under broad assumptions on $\kappa$ and $J$.Of special interest is the case when the order of the operator ${\mathcal L}^{\kappa}$ is smaller than or equal to 1. Our approach rests on imposingsuitable cancellation conditions on the internal drift coefficient$$\int_{r\leq |z| < 1} z \kappa(x,z)J(z)dz\,,\qquad 0 < r \leq 1\,,$$which allows us to handle the non-symmetry of$z\mapsto \kappa(x,z)J(z)$.The results are new even for the $1$-stable Lévy measure $J(z)=|z|^{-d-1}$.