The fundamental gap conjecture for Dirichlet eigenvalue problem on Euclidean domains is a well-known conjecture and was solved by Ben Andrews and Julie Clutterbuck in 2011. In contrast, there is a conjecture on the fundamental gap for Steklov eigenvalue problem posed by J. Escobar in 1999. In this talk we shall present our progress on Escobar's conjecture, namely, we confirm it for a large class of Riemannian manifolds, including Euclidean domains. We will also discuss new results on the comparison of Steklov spectrum and a related Laplacian spectrum as a by-product of our method. This is a joint work (arXiv:1907.07340) with Chao Xia (Xiamen University, China).