How does one perform the Mitzvah of eliminating Chametz? One must burn it or crumble it into tiny pieces and throw it to the wind or into the sea. The custom though is to burn it. Chametz which was thrown into a public trash receptacle (not in one's possession and before the time when Chametz becomes forbidden) need not be burned according to Halacha.  

"Chametz which Pesach has Passed Over"  

One who has kept Chametz in his possession on Pesach has nullified the positive commandment of "You shall eradicate leaven from your homes" in addition to transgressing the negative commandment of "Chametz shall not be seen with you." Our Sages have thus fined one who has transgressed this commandment by decreeing (Pesachim 28a): "Chametz owned by a Jew which was in his possession during Pesach is prohibited to benefit from." Such Chametz becomes prohibited to benefit from for the person who kept it in his possession on Pesach as well as for others. Even if others do not know that this Chametz was kept during Pesach, one who does know must notify others of this in order to prevent them from the prohibition of eating it.    

Sale of Chametz   

It is customary among the Jewish people, especially with regards to store, factory, and warehouse owners, to sell one's Chametz on Erev Pesach. The sale of Chametz can be carried out by signing a "power of attorney" appointing a Torah scholar to sell it which can be obtained from rabbis of synagogues all over the world. It is highly advised that one carry out this "Sale of Chametz" as is customary, especially for those who would like to keep Chametz which would be wasteful to dispose of in their possession, such as expensive alcoholic beverages and the like. 

The basis for the sale of Chametz is that Chametz owned by a non-Jew on Pesach does not become prohibited to benefit from after Pesach, for non-Jews are not commanded with regards to the prohibitions of Chametz on Pesach at all and they may keep it in their possession on Pesach. Thus, by selling one's Chametz by performing certain transactions prescribed by Halacha, the Chametz is sold to the non-Jew absolutely. 

How is this sale carried out? Rabbis all over the world meet with a non-Jew on Erev Pesach and explain to him clearly the contents of the contract which he will be signing on. The non-Jew then puts down a "down-payment" sometimes amounting to several hundreds of dollars for the Chametz being sold to him. It is agreed upon that after the holiday of Pesach, if he wishes he may pay the balance of his debt (which usually amounts to several millions of dollars based on the value of the Chametz which he has purchased), at which point he may go and collect all of the Chametz, wherever it may be. However, if he does not pay the balance, the Chametz shall return to its original Jewish owners who will then be permitted to eat it or sell it as they see fit.  

Although it is almost certain that the non-Jew will not come after Pesach and claim the Chametz, nevertheless, since he indeed has the ability to, the sale is valid. Similarly, Maran HaShulchan Aruch (Chapter 448) states: "If Chametz was sold or given as a gift to a non-Jew before Pesach, although it was sold to a non-Jew who will certainly not touch it at all, rather, he will safeguard it for him (the Jew) until after Pesach at which point he will give it back to him, this is permissible." The source for this ruling stems from the ruling of the Terumat Ha'Deshen regarding this matter.   

Where the Sold Chametz Should be Placed 

If one sells his Chametz to a non-Jew, one must conceal it in a designated closet or room and to mark the closet/room that it contains Chametz. By doing so, one will prevent mistaken consumption of these items during Pesach.   

Purchasing Chametz Items after Pesach 

One who is truly G-d-fearing should buy Chametz items after Pesach only from G-d-fearing store owners and grocers who have sold their Chametz to a non-Jew before Pesach via the local rabbinate or any other Kashrut organization, as is customary.