  Modified Newtonian dynamics (MOND) is a possible way to explain the flat
galaxy rotation curves without invoking the existence of dark matter. It is
however quite difficult to predict such a phenomenology in a consistent field
theory, free of instabilities and admitting a well-posed Cauchy problem. We
examine critically various proposals of the literature, and underline their
successes and failures both from the experimental and the field-theoretical
viewpoints. We exhibit new difficulties in both cases, and point out the hidden
fine tuning of some models. On the other hand, we show that several published
no-go theorems are based on hypotheses which may be unnecessary, so that the
space of possible models is a priori larger. We examine a new route to
reproduce the MOND physics, in which the field equations are particularly
simple outside matter. However, the analysis of the field equations within
matter (a crucial point which is often forgotten in the literature) exhibits a
deadly problem, namely that they do not remain always hyperbolic. Incidentally,
we prove that the same theoretical framework provides a stable and well-posed
model able to reproduce the Pioneer anomaly without spoiling any of the
precision tests of general relativity. Our conclusion is that all MOND-like
models proposed in the literature, including the new ones examined in this
paper, present serious difficulties: Not only they are unnaturally fine tuned,
but they also fail to reproduce some experimental facts or are unstable or
inconsistent as field theories. However, some frameworks, notably the
tensor-vector-scalar (TeVeS) one of Bekenstein and Sanders, seem more promising
than others, and our discussion underlines in which directions one should try
to improve them.
