In this paper a new necessary and sufficient condition is proposed for the solvability of the Disturbance Rejection by Measurement Feedback (DRMF) problem involving linear structured systems. The associated system graph can be used to easily check whether or not the condition holds. In relation to the DRMF problem, two issues related to the available sensor set have been investigated: - Sensor number and location: when the DRMF problem is not solvable with the given sensor set, how many sensors are needed and where should they be located to make the DRMF problem solvable? - Sensor classification: when the DRMF problem is solvable with the given sensor set, what is the impact of the failure of a sensor on the solvability? Based on the new condition, a lower bound can be determined for the number of sensors required. Moreover, measuring state variables outside a given subset is found to be of no use. To solve the DRMF problem, it is sufficient to measure state variables sufficiently close to the disturbances in the associated system graph. Sensor classification is used to distinguish between essential sensors, i.e. sensors for which failure leads to unsolvability, and useless sensors that have no impact on solvability. Partial classification results are provided for the general case and a complete characterization of essential and useless sensors is provided for the single disturbance case. (C) 2011 Elsevier Ltd. All rights reserved.