In Singapore, my shares are held by an organization known as the Central Depository (CDP). Regardless of which stock brokerage firm I am using, I can see the same shares, as all of the shares I own are held in the central depository. For example, if I have 5 brokerage accounts at brokers A, B, C, D, E, I will be able to buy shares through one account, and sell through the others, because all brokerage accounts can see the same shares. I can also transfer shares to another person by instructing the central depository to take shares from my account and deposit them into the other person's account.
In Malaysia, a similar system exists: the Central Depository System (CDS).
What is the name of the organization that acts as the central depository for shares in the US? If there is no such organization, where are shares held? How can I know if I really own the shares that the broker says I own, and that the broker is not cheating (brokers could just record fictitious trades and fictitious numbers of shares)?