  We study a tropical analogue of the projective dual variety of a
hypersurface. When $X$ is a curve in $\mathbb{P}^2$ or a surface in
$\mathbb{P}^3$, we provide an explicit description of $\text{Trop}(X^*)$ in
terms of $\text{Trop}(X)$, as long as $\text{Trop}(X)$ is smooth and satisfies
a mild genericity condition. As a consequence, when $X$ is a curve we describe
the transformation of Newton polygons under projective duality, and recover
classical formulas for the degree of a dual plane curve. For higher dimensional
hypersurfaces $X$, we give a partial description of $\text{Trop}(X^*)$.
