  We present new, practical algorithms for the hypersurface implicitization
problem: namely, given a parametric description (in terms of polynomials or
rational functions) of the hypersurface, find its implicit equation. Two of
them are for polynomial parametrizations: one algorithm, "ElimTH", has as main
step the computation of an elimination ideal via a \textit{truncated,
homogeneous} Gr\"obner basis. The other algorithm, "Direct", computes the
implicitization directly using an approach inspired by the generalized
Buchberger-M\"oller algorithm. Either may be used inside the third algorithm,
"RatPar", to deal with parametrizations by rational functions. Finally we show
how these algorithms can be used in a modular approach, algorithm
"ModImplicit", for avoiding the high costs of arithmetic with rational numbers.
We exhibit experimental timings to show the practical efficiency of our new
algorithms.
