We study Alexandrov curvature in the tropical projective torus with respect to the tropical metric. Alexandrov curvature is a generalization of classical Riemannian sectional curvature to more general metric spaces; it is determined by a comparison of triangles in an arbitrary metric space to corresponding triangles in Euclidean space. In the polyhedral setting of tropical geometry, triangles are a combinatorial object, which adds a combinatorial dimension to our analysis. We study the effect that the triangle types have on curvature, and what can be revealed about these types from the curvature. We find that positive, negative, zero, and undefined Alexandrov curvature can exist concurrently in tropical settings and that there is a tight connection between triangle combinatorial type and curvature. Our results are established both by proof and numerical experiments, and shed light on the intricate geometry of the tropical projective torus. This paper is dedicated to Bernd Sturmfels on the occasion of his 60th birthday.