Abstract In a randomized response (RR) survey on a sensitive character, the perception of sensitivity can vary from person to person and a section of the respondents may even be willing to disclose their true states rather than providing responses via the use of a randomization device. This opens up the prospect of allowing optional, rather than compulsory, RRs, leading to enhanced estimation efficiency. Research in this direction has received significant attention and the present chapter reviews these developments. We begin with the early work on qualitative characters and then pass on to more recent literature pertaining to quantitative characters. Results under both simple random sampling with replacement and general sampling designs are touched upon.