The problem of approximating a multivariate function by interpolatory functions is not an easy one. Many papers have already been published on the subject of polynomial interpolation and also on the subject of multivariate Padé approximation. But the problem of multivariate rational interpolation has only recently been considered. In this paper the authors generalize several methods of univariate rational interpolation to the multivariate case. Writing down the system of defining equations for the unknown numerator and denominator coefficients multivariate determinantal formulas are given for the rational interpolant. However, unlike the univariate case, the determinants cannot easily be computed recursively. Therefore, the authors present an Aitken-Neville-like algorithm: the recursive scheme developed generates rational interpolants for which the degree of numerator and denominator is fairly high for the number of interpolation conditions satisfied and combinatorial difficulties soon arise for large interpolation sets. The last approach presented is a generalization of univariate Thiele interpolating continued fractions; introducing multivariate inverse differences the authors construct interpolating branched continued fractions. Its convergents turn out to yield quite good numerical results.