This paper is dedicated to the study of shape optimization problems for the first eigenvalue of the elliptic operator with drift L = −∆+V (x)·∇ with Dirichlet boundary conditions, where V is a bounded vector field. In the first instance, we prove the existence of a principal eigenvalue λ1(Ω, V ) for a bounded quasi-open set Ω which enjoys similar properties to the case of open sets. Then, given m > 0 and τ ≥ 0, we show that the minimum of the following non-variational problemThe existence when V is fixed, as well as when V varies among all the vector fields which are the gradient of a Lipschitz function, are also proved.The second interest and main result of this paper is the regularity of the optimal shape Ω * solving the minimization problemwhere Φ is a given Lipschitz function on D. We prove that the optimal set Ω * is open and that its topological boundary ∂Ω * is composed of a regular part, which is locally the graph of a C 1,α function, and a singular part, which is empty if d < d * , discrete if d = d * and of locally finite, 6, 7} is the smallest dimension at which there exists a global solution to the one-phase free boundary problem with singularities. Moreover, if D is smooth, we prove that, for each x ∈ ∂Ω * ∩ ∂D, ∂Ω * is C 1,1/2 in a neighborhood of x.