This paper provides a comprehensive review and insight of recent trends in the field of random number generator (RNG) and physically unclonable function (PUF) circuits implemented using different types of emerging resistive non-volatile (NVM) memory devices. We present a detailed review of hybrid RNG/PUF implementations based on the use of (i) Spin-Transfer Torque (STT-MRAM), and (ii) metal-oxide based (OxRAM), NVM devices. Various approaches on Hybrid CMOS-NVM RNG/PUF circuits are considered, followed by a discussion on different nanoscale device phenomena. Certain nanoscale device phenomena (variability/stochasticity etc), which are otherwise undesirable for reliable memory and storage applications, form the basis for low power and highly scalable RNG/PUF circuits. Detailed qualitative comparison and benchmarking of all implementations is performed.