  In this paper, a spectral method based on conformal mappings is proposed to
solve Steklov eigenvalue problems and their related shape optimization problems
in two dimensions. To apply spectral methods, we first reformulate the Steklov
eigenvalue problem in the complex domain via conformal mappings. The
eigenfunctions are expanded in Fourier series so the discretization leads to an
eigenvalue problem for coefficients of Fourier series. For shape optimization
problem, we use the gradient ascent approach to find the optimal domain which
maximizes $k-$th Steklov eigenvalue with a fixed area for a given $k$. The
coefficients of Fourier series of mapping functions from a unit circle to
optimal domains are obtained for several different $k$
