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It is now well established that laser plasma acceleration (LPA) is an innovative and good candidate in the beam acceleration field. Relativistic beams are indeed produced up to several GeV but their quality remains to be demonstrated in the highly demanding case of free-electron lasers (FELs). Several experiments have already shown the feasibility of synchrotron radiation delivery based on LPA but free-electron lasing has still to be achieved. Since the quality of the LPA beam inside the undulator is the critical issue, any LPA-based FEL experiment requires a refined characterization of the beam properties along the transport line and of the photon beam at the undulator exit. This characterization relies on diagnostics which must be adapted to the LPA specificities. Here, the electron and photon diagnostics already used on LPAs and required for LPA-based FELs are reviewed, and the critical points are illustrated using recent experiments performed around the world. [ABSTRACT FROM AUTHOR]