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Hilbert's 16th Problem, about the maximum number of limit cycles of planarpolynomial vector fields of a given degree $m$, has been one of the mostimportant driving forces for new developments in the qualitative theory ofvector fields. Increasing the dimension, one cannot expect the existence of afinite upper bound for the number of limit cycles of, for instance, $3$Dpolynomial vector fields of a given degree $m$. Here, as an extension of such aproblem in the $3$D space, we investigate the number of isolated invariant toriin $3$D polynomial vector fields. In this context, given a natural number $m$,we denote by $N(m)$ the upper bound for the number of isolated invariant toriof $3$D polynomial vector fields of degree $m$. Based on a recently developedaveraging method for detecting invariant tori, our first main result provides amechanism for constructing $3$D differential vector fields with a number $H$ ofnormally hyperbolic invariant tori from a given planar differential vectorfield with $H$ hyperbolic limit cycles. The strength of our mechanism instudying the number $N(m)$ lies in the fact that the constructed $3$Ddifferential vector field is polynomial provided that the given planardifferential vector field is polynomial. Accordingly, our second main resultestablishes a lower bound for $N(m)$ in terms of lower bounds for the number ofhyperbolic limit cycles of planar polynomial vector fields of degree $[m/2]-1$.Based on this last result, we apply a methodology due to Christopher & Lloyd toshow that $N(m)$ grows as fast as $m^3/128$. Finally, the above-mentionedproblem is also formulated for higher dimensional polynomial vector fields.

A version of Hilbert's 16th Problem for 3D polynomial vector fields: Counting isolated invariant tori