According to Euler’s theorem, for relatively prime numbers \(a\in \mathbb{Z}\) and \(n\in \mathbb{N}\), it holds that \(a^{\varphi(n)}\equiv 1 \pmod n\). The smallest natural number \(d\) with the property \(a^d\equiv1 \pmod n\) is called the order of \(a\) modulo \(n\). If \(d=\varphi(n) \), then \(a\) is called a primitive root modulo \(n\) and the set \(\{a^1, a^2, \ldots, a^{\varphi(n)}\} \) forms a reduced system of residues modulo \(n\). This implies that \(\mathbb{Z}_n^*\) is a cyclic multiplicative group, where the primitive root \(a\) is its generator. A primitive root modulo \(n\) does not exist for every natural number \(n\). It is shown that a primitive root modulo \(n\) exists if and only if \(n=2, 4, p^k\) or \(n=2p^k\), where \( p\) is a prime number and \(k\in \mathbb{N}\). Considering that \(\{a^0, a^1, \ldots, a^{\varphi(n)-2}\}\) is a reduced system of residues modulo \(n\), it leads to a definition of the index (discrete logarithm). If \(y\in\mathbb{Z}\) is relatively prime to \(n\), then the index of \(y\) to the base a modulo \(n\) equals \(x\in \{0, 1, \ldots, \varphi(n)-1\}\) such that \(y\equiv a^x\pmod n\). Indices have many properties similar those of logarithms where equalities are replaced with congruences modulo \(\varphi(n) \). In this graduate thesis, some applications of primitive roots and discrete logarithms are described, such as solving polynomial and exponential congruences, primality testing, and the key exchange protocol which is used in cryptography.