A metric is said to be (globally) median,  if any three points have a unique “median” which  lies  between  any  two  points  from  the  triple.  
Such  spaces  arise  naturally  in  many different contexts.  The property of being locally median can be viewed as a kind of
non-positive curvature condition.  We show that a complete uniformly locally median space is
globally median if and only if it is simply connected.  This is an analogue of the well known Cartan-Hadamard Theorem for non-positively curved manifolds, or more generally CAT(0) spaces.  However it leaves open a number of interesting questions.