An n×n real matrix is said to be totally positive if every minor is non-negative. In this paper, we are interested in totally positive completion problems, that is, does a partial totally positive matrix have a totally positive matrix completion? This problem has, in general, a negative answer. Therefore, we analyze the question: for which labeled graphs G does every partial totally positive matrix, whose associated graph is G, have a totally positive completion? When G is a path or a cycle, we give necessary and sufficient conditions that guarantee the existence of the desired completion.