In The Force Awakens, the Millennium Falcon starts out in the hands of Unkar Plutt, who stole it from the Irving Boys, who stole it from Gannis Ducain, who stole it from Han Solo, who originally got it from Lando Calrissian.
Because our understanding of the laws of property under the Empire, New Republic, and First Order are so lacking, for the sake of this analysis we’ll assume that U.S. law applies.
The Millennium Falcon makes a brief appearance in Revenge of the Sith, but we don’t know who owns it at that point. In the absence of any evidence to the contrary we’ll assume Lando Calrissian is the first lawful owner of the Millennium Falcon. At one point he engaged in a “game of chance”, defined as “poker, craps, roulette, or other game in which a player gives anything of value in the hope of gain, the outcome of which is determined largely by chance….” (Ohio Revised Code.2915.01) (Hereafter O.R.C). He lost that game of chance, in this case Sabaac, and ownership passed to Han Solo. Or did it?
If the game was not lawfully sanctioned, it would be an illegal debt that is unenforceable under U.S. common law so the Millennium Falcon would still belong to Lando Calrissian. Hence, the comment, “What have you done to my ship?”
If the game was legally sanctioned and the Millennium Falcon was used as payment, or partial payment, for a debt lawfully owed, then ownership would pass to Han Solo. (O.R.C. 2915.02 does not prohibit conduct in connection with gambling expressly permitted by law.) Thus the rejoinder by Han Solo to the above comment, “Your ship? Hey, remember you lost her to me, fair and square” asserting that ownership had passed to him.
The Millennium Falcon is considered personal property. For most personal property ownership is often determined by possession, even if the ownership, possession or interest is unlawful. (O.R.C. 2913.01) Because of this Han Solo either has an ultimate claim to the Falcon, having received it as a payment for a lawful debt, or can claim a superior interest second only to that of Lando Calrissian, since his possession is only one step removed from the lawful owner.
The Millennium Falcon is later stolen from Han Solo by Ducain. This allows Ducain to exert a possessory interest over the Millennium Falcon, even though his claim was unlawful.
And it was stolen from Ducain by the Irving Boys, who could then claim a possessory interest over the Millennium Falcon, even though their claim was unlawful.
And it was stolen from them by Unkar Plutt, who could also exert a possessory interest over the Millennium Falcon even though his claim was unlawful.
And it was stolen from Unkar by Rey and Finn, who could also claim a possessory interest over the Millennium Falcon even though their claim was unlawful as well.
According to the common law to determine the superior claim you simply work your way back up the chain. So Rey & Finn had a superior interest to everyone but Unkar’s, the Irving boys, Ducain, and Han Solo. Unkar’s interest trumps Rey & Finns. The Irving Boys have an interest that trumps Unkar’s. Ducains’ interest trumps the Irving Boys. And Han Solo has a superior claim to them all. And he then messes things up by dying. So now who owns the Millennium Falcon?
Well, assuming Han Solo died without a will, which seems pretty likely, who owns it will be determined by the laws of intestacy. Han Solo was married to Leia Organa. They were separated, but don’t seem to have made it a formal separation as recognized by the law. They had at least one child (Ben/Kylo Ren) but, as far as we know, neither of them had any children outside of the marriage. As a result, where there is a marriage and all the children are born of that marriage then the whole goes to the surviving spouse. (O.R.C. 2105.06.) So even though Rey may be flying the ship, Leia inherited all rights to the Millennium Falcon and it ultimately belongs to her… assuming Han won it in a lawful game of Sabaac…but he was a great swindler and never one for the rules, so it’s entirely possible it still belongs to Lando.