  A Taylor variety consists of all fixed order Taylor polynomials of rational
functions, where the number of variables and degrees of numerators and
denominators are fixed. In one variable, Taylor varieties are given by rank
constraints on Hankel matrices. Inversion of the natural parametrization is
known as Pad\'e approximation. We study the dimension and defining ideals of
Taylor varieties. Taylor hypersurfaces are interesting for projective geometry,
since their Hessians tend to vanish. In three and more variables, there exist
defective Taylor varieties whose dimension is smaller than the number of
parameters. We explain this with Fr\"oberg's Conjecture in commutative algebra.
