Taylor's Theorem in Multiple Dimensions

What do higher order derivatives look like for functions that map vector spaces of multiple dimensions to other vector spaces of multiple dimensions? Well if the first derivative at a point is a linear operator, then the second derivative at a point is a bilinear operator, and the third a trilinear operator - and on and on. How can we assemble these derivatives into a Taylor polynomial approximation of a function, and how good of an approximation is it?