The transverse phase space of a beam in an accelerator can be characterized using well-established methods based on observation of changes in the beam profile between screens at different locations along the beamline, or on observation of changes on a single screen for different strengths of upstream quadrupoles. Studies on CLARA FE (the Compact Linear Accelerator for Research and Applications Front End, at Daresbury Laboratory, UK) show that where the beam has a complicated (nonelliptical) distribution in transverse phase space, conventional analysis techniques aimed at characterizing the beam in terms of the emittances and Courant–Snyder parameters fail to provide a good description of the beam behavior. Phase space tomography, however, allows the construction of a detailed representation of the phase space distribution that provides a better model for the beam. We compare the results from three measurement and analysis techniques applied on CLARA FE: emittance and optics measurements from observations on three screens; emittance and optics measurements from quadrupole scans on a single screen; and phase space tomography. The results show the advantages of phase space tomography in providing a detailed model of the beam distribution in phase space. We present the first experimental results from four-dimensional phase space tomography, which gives an insight into beam properties (such as the transverse coupling) that are of importance for optimizing machine performance.