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Abstract: Monadic least fixed point logic MLFP is a natural logic whose expressiveness lies between that of first-order logic FO and monadic second-order logic MSO. In this paper, we take a closer look at the expressive power of MLFP. Our results are: [(1)] MLFP can describe graph properties beyond any fixed level of the monadic second-order quantifier alternation hierarchy. [(2)] On strings with built-in addition, MLFP can describe at least all languages that belong to the linear time complexity class DLIN. [(3)] Settling the question whether or, equivalently, settling the question whether would solve open problems in complexity theory: “” would imply that whereas “” would imply that . Apart from this we give a self-contained proof of the previously known result that MLFP is strictly less expressive than MSO on the class of finite graphs. [Copyright &y& Elsevier]