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