The book %LorentzR92a, on multivariate Birkhoff interpolation, provides a good survey.

For recent work, look for the papers of Len Bos, Mariano Gasca & Maeztu, Hakop Hakopian & Sahakian, G.G. Lorentz & R.A. Lorentz, Thomas Sauer (& Yuan Xu), Ian Sloan, e.g., in the spline bibliography.

Here is a very recent bibliography (in postscript) on multivariate polynomial interpolation, which will appear as part of a survey, by Gasca and Sauer, in a special issue of

- Interpolation from spaces spanned by monomials
- Interpolation by radial polynomials
- the least interpolant
- Ideal interpolation
- error in linear interpolation at the vertices of a simplex.
- Sloan's hyperinterpolation
- (history), i.e., a place to hold facts before there is an appropriate place to put them
- Padua points of Marco Caliari, Stefano De Marchi, and Marco Vianello.

- Hesthaven98 has near-optimal interpolation points in a triangle.