In this paper, we propose a new approach to design globally convergent reduced-order observers for nonlinear control systems via contraction analysis and convex optimization. Despite the fact that contraction is a concept naturally tailored for state estimation, the existing solutions are either local or relatively conservative when applying to physical systems. We show that this problem can be translated into an off-line search for a coordinate transformation after which the dynamics is contracting. The obtained sufficient condition consists of some easily verifiable differential inequalities, which, on one hand, identify a very general class of “detectable” nonlinear systems, and on the other hand, can be expressed as computationally efficient convex optimization, making the design procedure systematic. Finally, we illustrate the proposed method with a numerical example.