  In this paper, the discontinuous Petrov--Galerkin approximation of the
Laplace eigenvalue problem is discussed. We consider in particular the primal
and ultra weak formulations of the problem and prove the convergence together
with a priori error estimates. Moreover, we propose two possible error
estimators and perform the corresponding a posteriori error analysis. The
theoretical results are confirmed numerically and it is shown that the error
estimators can be used to design an optimally convergent adaptive scheme.
