We prove Schur positivity conjectures due to Fomin-Fulton-Li-Poon and Okounkov and give several related results. We show that Schur polynomials satisfy an interesting Schur-log-concavity property. We discuss a "higher Klyachko problem" about inequalities between Littlewood-Richardson coefficients and formulate a general Schur positivity conjecture involving certain mysterious convex polytopes. The talk is based on a joint work with Thomas Lam and Pavlo Pylyavskyy.