We study a tropical analogue of the projective dual variety of a hypersurface. When @math is a curve in @math or a surface in @math , we provide an explicit description of @math in terms of @math , as long as @math is smooth and satisfies a mild genericity condition. As a consequence, when @math 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 @math , we give a partial description of @math .
 Finding the so-called characteristic numbers of the complex projective plane @math is a classical problem of enumerative geometry posed by Zeuthen more than a century ago. For a given @math and @math one has to find the number of degree @math genus @math curves that pass through a certain generic configuration of points and in the same time are tangent to certain generic configuration of lines. The total number of points and lines in these two configurations is @math so that the answer is a finite integer number. In this paper we translate this classical problem to the corresponding enumerative problem of tropical geometry in the case when @math . Namely, we show that the tropical problem is well-posed and establish a special case of the correspondence theorem that ensures that the corresponding tropical and classical numbers coincide. Then we use the floor diagram calculus to reduce the problem to pure combinatorics. As a consequence, we compute genus 0 characteristic numbers of @math in terms of open Hurwitz numbers. Duality of curves is one of the important aspects of the classical'' algebraic geometry. In this paper, using this foundation, the duality of tropical polynomials is constructed to introduce the duality of Non-Archimedean curves. Using the development of mechanism'' which is based on distortion'' values and their matrices, we discuss some aspects refereing to quadrics with respect to their dual objects. This topic includes also the induced dual subdivision of Newton Polytope and its compatible properties. Finally, a regularity of tropical curves in the duality sense is generally defined and, studied for families of tropical quadrics. We study smooth tropical plane quartic curves and show that they satisfy certain properties analogous to (but also different from) smooth plane quartics in algebraic geometry. For example, we show that every such curve admits either infinitely many or exactly 7 bitangent lines. We also prove that a smooth tropical plane quartic curve cannot be hyperelliptic.