The stateful public-key encryption scheme proposed by Bellare, Kohno and Shoup in 2006 significantly improves the efficiency of the encryption operation of ElGamal-like public-key encryption schemes. The basic idea of the stateful public-key encryption scheme is to reuse some random parameters in the encryption algorithm by maintaining a state to save the current random variable, which is used to generate the random parameters. This turns out to be highly effective in reducing heavy computations like exponentiation operations in the encryption process. Since its invention, several variants and extensions of the stateful public key encryption scheme have been proposed. This chapter provides an extensive survey of original stateful public-key encryption scheme and their extensions. Also, possible applications of stateful encryption schemes for building up lightweight asymmetric encryption primitives for the Internet of things (IoT) environment are discussed.