The main objects of interest in this thesis are elliptic curves and the objects related to them. We will mostly investigate the isogenies of elliptic curves over number fields of small degree, but we will also give some interesting results about the torsion of rational elliptic curves when the torsion is considered over some specific cyclotomic fields. Let \(E/\mathbb{Q}\) be an elliptic curve and \(p \leq 11\) a prime. We give a complete classification of the possibilities for \(E(\mathbb{Q}(\zeta_p))\) as well as for \(E(\mathbb{Q}(\zeta_{16}))\) and \(E(\mathbb{Q}(\zeta_{27}))\). Using the previous result of Gužvić and Krijan [41], we are able to give a complete classification for \(E(\mathbb{Q}(\mu_{p^\infty}))_{tors}\). Here, the set \(\mu_{p^\infty}\) is the set of all complex numbers \(\omega\) for which there exists non-negative integer \(k\) such that \(\omega^{p^k} =1\). Moving on to the isogenies, we can ask ourselves which cyclic isogeny degrees are possible for a non-CM elliptic curve \(E/\mathbb{Q}\) if the isogeny is defined over a low degree number field. Since the presence of a (cyclic) isogeny is invariant under quadratic twisting, we actually determine which cyclic isogeny degrees are possible for a non-CM elliptic curve defined over a quadratic field \(K\), but which has a rational \(j\)-invariant. Similarly, given a quadratic field \(K\) and an elliptic curve \(E/K\), we can ask ourselves which cyclic isogeny degrees are possible for \(E\). We use the fact that the pairs \((E/K,C)\), where \(C\) is a cyclic subgroup of \(E\) defined over \(K\), are parametrized by \(K\)-rational points on the modular curve \(X_0(n)\). Hence, we should look for quadratic points on curves \(X_0(n)\). We are able to determine all the quadratic points on all bielliptic curves \(X_0(n)\) for which this has not been done before. This covers the cases \(n = 60,62,69,79,83,89,92,94,95,101, 119,131\). Our proof relies a lot on the relative symmetric Chabauty’s method developed by Siksek [82] and used by Box [13] on a related problem. We also make some improvements to the method, both from the computational and algebraic perspective. The Magma [12] code which verifies our computations can be found on the links given at the beginning of each corresponding chapter.