Given a nodal curve $X$, we introduce the notion of a good compactified Jacobian as a proper algebraic space which is the good moduli space of an open connected substack of the stack $\Coh$ of rank 1 torsion-free sheaves on $X$.

After providing a local description of the stack $\Coh$ for every nodal curve $X$, we focus on necklaces of genus $g=1$, namely nodal curves whose dual graph is an $n$-agon. 

We first give a geometric and combinatorial classification of all good compactified Jacobians of a fixed necklace $X$. 

Subsequently we consider all the stable necklaces in $\overline{\mathcal{M}}_{1,n}$ and we give a combinatorial classification of all the possible choices of a good compactified Jacobian for every necklace in $\overline{\mathcal{M}}_{1,n}$, such that this choice is compatible with degenerations of curves. 

In the last part of this thesis, we classify all the good compactified Jacobians of a vine curve with $3$ nodes, namely a curve given by two copies of $\mathbb{P}^1$ meeting transversally at $3$ nodes.