In this thesis we will classify the possible torsion structures of elliptic curves with rational \(j\)-invariant defined over number fields. We start with elliptic curves defined over \(\mathbb{Q}\). Let \(K\) be a sextic number field. We determine all the possibilities \(G\) for \(E(K)_{tors}\) and we prove that for each such possible group \(G\), with the exception of the group \(C_3 \bigoplus C_{18}\), that there exist an elliptic curve \(E / \mathbb{Q}\) and a sextic number field \(K\) such that \(E(K)_{tors} \cong G\). Additionally, we provide a partial result regarding the group \(C_3 \bigoplus C_{18}\). For a positive integer \(d\), define \(\Phi(d)\) to be the set of possible isomorphism classes of groups \(E(K)_{tors}\), where \(K\) runs through all number fields \(K\) of degree \(d\) and \(E\) runs through all elliptic curves over \(K\). For a positive integer \(d\), define \(\Phi_{\mathbb{Q}}(d)\) to be the set of possible isomorphism classes of groups \(E(K)_{tors}\), where \(K\) runs through all number fields \(K\) of degree \(d\) and \(E\) runs through all elliptic curves over \(\mathbb{Q}\). Define \(\Phi_{j \in \mathbb{Q}}(d)\) to be the set of possible isomorphism classes of groups \(E(K)_{tors}\), where \(K\) runs through all number fields \(K\) of degree \(d\) and \(E\) runs through all elliptic curves over \(K\) with \(j(E) \in \mathbb{Q}\). With the help of the previously mentioned result, we are able to completely determine the sets \(\Phi_{j \in \mathbb{Q}}(p)\), where \(p\) is a prime number. More precisely, our result is the following. Let \(K\) be a number field such that \([K : Q] = p\) and \(E / K\) an elliptic curve with rational \(j\)-invariant. The following holds: 1. If \(p \geq 7\), then \(E(K)_{tors} \in \Phi(1)\). 2. If \(p = 3\) or \(p = 5\), then \(E(K)_{tors} \in \Phi_{\mathbb{Q}}(p)\). 3. If \(p = 2\), then \(E(K)_{tors} \in \Phi_{\mathbb{Q}}(2)\) or \(E(K)_{tors} \cong \mathbb{Z}/13\mathbb{Z}\). In the sixth chapter, we are able to determine all the sets \(\Phi_{\mathbb{Q}}(pq)\), where \(p\) and \(q\) are prime numbers. Most of these cases follow easily from previously known results and the results in the first two chapters of this thesis. In most cases we have \(\Phi_{\mathbb{Q}}(pq) = \Phi_{\mathbb{Q}}(p) \cup \Phi_{\mathbb{Q}}(q)\). A detailed description of the sets \(\Phi_{\mathbb{Q}}(pq)\) can be found in the fifth chapter of this thesis. Some of the proofs in the thesis rely on extensive computations in Magma [3]. All of the programs and calculations used for the proofs can be found in the last chapter.