  Let $K$ be a number field, and let $E/K$ be an elliptic curve over $K$. The
Mordell--Weil theorem asserts that the $K$-rational points $E(K)$ of $E$ form a
finitely generated abelian group. In this work, we complete the classification
of the finite groups which appear as the torsion subgroup of $E(K)$ for $K$ a
cubic number field.
  To do so, we determine the cubic points on the modular curves $X_1(N)$ for
\[N = 21, 22, 24, 25, 26, 28, 30, 32, 33, 35, 36, 39, 45, 65, 121.\] As part of
our analysis, we determine the complete list of $N$ for which $J_0(N)$ (resp.,
$J_1(N)$, resp., $J_1(2,2N)$) has rank 0. We also provide evidence to a
generalized version of a conjecture of Conrad, Edixhoven, and Stein by proving
that the torsion on $J_1(N)(\mathbb{Q})$ is generated by
$\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$-orbits of cusps of
$X_1(N)_{\bar{\mathbb{Q}}}$ for $N\leq 55$, $N \neq 54$.
