The Casimir effect requires a complete axiomatic description in terms of the algebraic approach to quantum field theory. When the confinement of a field between two parallel plates is described as a boundary condition on the field configurations, the dynamics of the underlying system cannot be described in terms of a initial value problem on globally hyperbolic spacetimes. This calls on the one hand for an ad hoc construction of the field algebra, which partially recovers the usual one, based on a causal propagator. On the other hand, an adapted definition of the Hadamard property is needed, in order to recover a coherent regularization scheme and all known expectation values of the energy, characterizing the Casimir effect. In this talk, all these issues will be addressed.