Censors in the Roman Republican government were highly esteemed and had to be former Consuls. They maintained the role of the Senate, deciding who was morally fit to sit on the Senate. They also conducted the census of Roman citizens for conscription into the army. This office was regarded as the crowning of Roman civic life, and in a sense the culminating achievement of a political career.
Its powers were very extensive and they included the right to inquire into the lives of citizens and punish any tendency to indulge in immoral habits that departed from the traditional and established way of living.
It is difficult to say what power, if any they may have had in the imperial system as this sort of supervision was really at the whim of the Emperor himself. They may have still maintained the census, but doubtful that they did much else.
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After the Sullan reforms of 81 BC new senators were enrolled automatically, much reducing the influence of the censors over membership in the Senate.