An intriguing prediction of quantum mechanics is that the vacuum is not empty. In fact, quantum theory predicts that it teems with virtual particles fleeting in and out of existence. These vacuum fluctuations have several measurable consequences, producing for example the Lamb shift and Casimir forces. The dynamical Casimir effect (DCE) is an example of a dynamical quantum vacuum phenomena, where a nonadiabatic time-dependent boundary condition - for example a relativistic accelerating mirror - can amplify vacuum fluctuations and produce real observable photons. In this talk I will present a theoretical proposal for observing the DCE in a quantum mechanical superconducting waveguide, with emphasis on photon statistics and correlations that can be used as a signatures in experimental measurements of the effect. I will also briefly present experimental data from measurements of the dynamical Casimir radiation in a superconducting microwave circuit.