We consider observer designs for an exhaust-gas turbocharger based on a third order nonlinear model of the air path of a diesel engine. To propose an observer, the system is analyzed in view of different observability properties for nonlinear systems. We derive explicit constraints for the system state guaranteeing observability. Based on that we design a high-gain observer and a sliding mode observer for the system. The performance of both designs has been assessed through a simulation study.