The paper addresses the question when can an abstract algebraic curve be given by a nondegenerate polynomial in 2 variables. A polynomial is called nondegenerate if its zero set has singularities except those that can be predicted by its Newton polytope. Over an algebraically closed field, any curve of genus at most 4 can be modeled in this way. For genus \(g\geq 4\), the moduli space of curves that can be modeled in this way is equal to \(2g+1\), except for \(g=7\), where the dimension is 16. The paper also gives results for non-closed fields.