Two Latin squares of order are r-orthogonal if their superposition produces exactly r distinct ordered pairs. If the second square is the transpose of the first one, we say that the first square is r-self-orthogonal, denoted by r-SOLS. It has been proved that for any integer , there exists an r-SOLS if and only if and . In this paper, we give an almost complete solution for the existence of r-self-orthogonal Latin squares.