In this talk, we discuss the concept of stable extremal domains for the first Dirichlet eigenvalue of the Laplacian operator. We classify the stable extremal domains in the 2-sphere and higher-dimensional spheres when the boundary is minimal. Additionally, we establish topological bounds for stable domains in general compact Riemannian surfaces, assuming either nonnegative total Gaussian curvature or small volume. This is a joint work with Marcos P. Cavalcante (UFAL, Brazil).