abstract: We study the simplex M1(B) of probability measures on a Bratteli diagram B which are invariant with respect to the tail equivalence relation. Equivalently, M1(B) is formed by probability measures invariant with respect to a homeomor- phism of a Cantor set. We prove a criterion of unique ergodicity of a Bratteli diagram. In the case of a finite rank k Bratteli diagram B, we give a criterion for B to have exactly 1 ≤ l ≤ k ergodic invariant measures and describe the structures of the diagram and the subdiagrams which support these measures. We also find sufficient conditions under which a Bratteli diagram of arbitrary rank has a prescribed number (finite or infinite) of probability ergodic invariant measures. This is a joint work with S. Bezuglyi and J. Kwiatkowski.