A Lie algebra \(\mathfrak{g}\) is called a Frobenius Lie algebra provided that there is a linear form \(l\in \mathfrak{g}^*\) whose stabilizer with respect to the coadjoint representation of \(\mathfrak{g}\) is trivial. Frobenius Lie algebras are always even dimensional. There is a unique 2-dimensional Frobenius Lie algebra, namely the Lie algebra of the group of affine transformations of the line. 4-dimensional Frobenius Lie algebras over an algebraically closed field of characteristic zero are known [see \textit{A. I. Ooms}, J. Algebra, 32, 488--500 (1974; Zbl 0355.17014)]. In the present paper the authors classify Frobenius Lie algebras of dimension 4 over an arbitrary field of characteristic \(\neq 2\), and Frobenius Lie algebras of dimension 6 over an algebraically closed field of characteristic zero.