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To every singular reduced projective curve one can associate many fine compactified Jacobians, depending on the choice of a polarization on , each of which yields a modular compactification of a disjoint union of a finite number of copies of the generalized Jacobian of . We investigate the geometric properties of fine compactified Jacobians focusing on curves having locally planar singularities. We give examples of nodal curves admitting nonisomorphic (and even nonhomeomorphic over the field of complex numbers) fine compactified Jacobians. We study universal fine compactified Jacobians, which are relative fine compactified Jacobians over the semiuniversal deformation space of the curve . Finally, we investigate the existence of twisted Abel maps with values in suitable fine compactified Jacobians. References