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The Mutual Fund Theorem (MFT) is considered in a general semimartingalefinancial market S with a finite time horizon T, where agents maximize expectedutility of terminal wealth. It is established that:  1) Let N be the wealth process of the num\'eraire portfolio (i.e. the optimalportfolio for the log utility). If any path-independent option with maturity Twritten on the num\'eraire portfolio can be replicated by trading \emph{only}in N, then the (MFT) holds true for general utility functions, and thenum\'eraire portfolio may serve as mutual fund. This generalizes Merton'sclassical result on Black-Scholes markets.  Conversely, under a supplementary weak completeness assumption, we show thatthe validity of the (MFT) for general utility functions implies the samereplicability property for options on the num\'eraire portfolio describedabove.  2) If for a given class of utility functions (i.e. investors) the (MFT) holdstrue in all complete Brownian financial markets S, then all investors use thesame utility function U, which must be of HARA type. This is a result in thespirit of the classical work by Cass and Stiglitz.

In which Financial Markets do Mutual Fund Theorems hold true?