A pump rod clamp for gripping a pump rod in an oil, gas or water well. The pump rod clamp comprises a central housing having a longitudinal bore for receiving a pump rod, a plurality of clamping members receivable within the central housing, and one or more actuators to cause the clamping members to be moved between an activated position where they grippingly engage the pump rod and a deactivated position where they are released from contact with the pump rod. Each of the clamping members have a leading face that includes one or more gripping inserts releasably secured thereto. The gripping inserts having outer faces generally aligned with the pump rod for grippingly engaging the pump rod when the clamping members are in their activated position such that force exerted by each individual clamping member against the pump rod is concentrated and distributed over the outer faces of the one or more gripping inserts of each clamping member.