The celebrated "Hasse Principle" holds for plane conics over a
global field, but generally not for algebraic curves of positive genus.
 Isolated examples of curves violating the Hasse Principle go back to
Lind, Reichardt and Selmer in the 1940s and 1950s.  Many more examples
have been found since, and it now seems likely that the Hasse principle
should, in some suitable sense, most often be false.  However it is
challenging to make, let alone prove, a precise statement to this effect.

In this talk I will discuss certain "anti-Hasse principles",
some which are conjectural and others
(more modest) which are known to hold.  Some attention will be devoted
to algorithmic considerations and to the function field case.