Let \({\mathcal T}_ c^ n\) and \({\mathcal T}_ r^ n\) be the linear spaces of complex symmetric and real symmetric Toeplitz matrices, respectively. The inverse eigenvalue problem for symmetric complex-valued Toeplitz matrices (IEPSCTM) consists of finding \(T\in {\mathcal T}_ c^ n\) with prescribed \(n\) complex eigenvalues \(\{\sigma_ 1,\dots,\sigma_ n\}\). The inverse eigenvalue problem for symmetric real-valued Toeplitz matrices (IEPSRTM) consists of finding \(T\in{\mathcal T}_ r^ n\) with prescribed \(n\) real eigenvalues \(\{\sigma_ 1,\dots,\sigma_ n\}\). The author outlines approaches to the IEPSCTM and IEPSRTM using methods of complex and real algebraic geometry. In particular on the base of Bézout's theorem he shows that for \(n\leq 4\) the IEPSCTM is always solvable and the number of distinct Toeplitz matrices with a prescribed spectrum is \(n!\) counted with multiplicities. For the IEPSRTM he develops the following method. Let \(\sigma_ 1=\sigma_ 1(T)\leq\dots\leq \sigma_ n=\sigma_ n(T)\), \(\sigma=\sigma(T)=(\sigma_ 1,\dots,\sigma_ n)\) be the eigenvalues of \(T\in {\mathcal T}_ r^ n\) arranged in an increasing order. Let \({\mathcal T}_{r,0}^ m\subset {\mathcal T}_ r^ n\) be the subset of all matrices with pairwise distinct eigenvalues and \(K_ i^ n\), \(i=1,\dots,\kappa_ n\) be the connected components of \({\mathcal T}_{r,0}^ n\). Then he defines the map \(\sigma: K_ i^ n \to \Lambda_ 0^ n\), where \(\Lambda_ 0^ n=\{x\), \(x=(x_ 1,\dots,x_ n)\), \(x_ 1