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We review the interplay of fuzzy field theories and matrix models, with anemphasis on the phase structure of fuzzy scalar field theories. We give aself-contained introduction to these topics and give the details concerning thesaddle point approach for the usual single trace and multitrace matrix models.We then review the attempts to explain the phase structure of the fuzzy fieldtheory using a corresponding random matrix ensemble, showing the strength andweaknesses of this approach. We conclude with a list of challenges one needs toovercome and the most interesting open problems one can try to solve.

Phase strucutre of fuzzy field theories and multitrace matrix models