A locally testable language L is a language with the property that for some nonnegative integer k, called the order of local testability, whether or not a word u is in the language L depends on (1) the preΓΏx and su x of the word u of length k -1 and (2) the set of subwords of length k of the word u. For given k the language is called k-testable. We improve the upper bound on the order of local testability of a locally testable deterministic ΓΏnite automaton with n states to 1 2 (n 2 -n) + 1. This bound is the best possible. We give an answer to the following conjecture of Kim, McNaughton and McCloskey for deterministic ΓΏnite locally testable automata with n states: "Is the order of local testability no greater than O(n 1:5 ) when the alphabet size is two?" Our answer is negative. In the case of size two the situation is the same as in general case: the order of local testability is (n 2 ). The necessary and su cient conditions for the language of an automaton to be k-testable are given in terms of the length of paths of a related graph. Some estimates of the bounds on the order of local testability follow from these results.