  A regular set of words is ($k$-)locally testable if membership of a word in
the set is determined by the nature of its subwords of some bounded length $k$.
In this article we study groups for which the set of all geodesic words with
respect to some generating set is ($k$-)locally testable, and we call such
groups ($k$-)locally testable. We show that a group is \klt{1} if and only if
it is free abelian. We show that the class of ($k$-)locally testable groups is
closed under taking finite direct products. We show also that a locally
testable group has finitely many conjugacy classes of torsion elements.
  Our work involved computer investigations of specific groups, for which
purpose we implemented an algorithm in \GAP\ to compute a finite state
automaton with language equal to the set of all geodesics of a group (assuming
that this language is regular), starting from a shortlex automatic structure.
We provide a brief description of that algorithm.
