This book explores the different approaches towards proving Artin's 'primitive root' conjecture unconditionally and the elliptic curve analogue of the same. The conjecture was posed by E. Artin in 1927, and it still remains an open problem. In 1967, C. Hooley proved the conjecture based on the assumption of the generalized Riemann hypothesis. Thereafter, the mathematicians tried to get rid of the assumption and it seemed quite a daunting task. The best results we have so far are by R. Gupta and M. Ram Murty (1983), and D. R. Heath-Brown (1986). It states that there can be at most 12 exceptional integers or at most 2 exceptional primes for which the conjecture does not hold. But the question - 'Which ones?', remains illusive. The first part of this book deals with these sieve theoretic results by Gupta-Murty and Heath-Brown. The second half discusses the elliptic curve analogue of the conjecture which was proposed by Lang and Trotter in 1977 and was proved by Gupta and Murty in 1986, assuming the generalized Riemann hypothesis, for curves with complex multiplication. These discussions will help the reader gain a broad perspective of the conjecture and the current advancements.

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