A principal way the ultra-rich and corporations pay less taxes is by shifting where they declare income to places like Caribbean islands with a very low or non-existent tax rate.  The New York Times has run an excellent series of articles describing how Wilber Ross, the Commerce Secretary, Apple and others have benefited, summarized in the following article.

Paradise Papers shine light on where the elite keep their money Michael Forsythe New York Times November 5, 2017

Tax havens are also used to cover up bribery, stealing government assets and other illegal transactions.

Paradise Papers reveal how tax havens damage Africa  Kate Hairsine DW November 11, 2017

The Paradise Papers hacking and the consequences of privacy Jake Bernstein New York Times November 7, 2017

…a lawyer at Mossack Fonseca candidly wrote in a confidential internal memorandum, “95 percent of our work” is “selling vehicles to avoid paying taxes.” The amounts involved are staggering. An estimated 8 percent of household financial wealth is held offshore, representing a loss in annual global tax revenue of about $190 billion. But this pales in comparison to the tax avoidance and tax evasion by the large multinational companies that use this system. All told, more than $7.6 trillion may well be hidden in tax havens around the world, according to Gabriel Zucman, an economist at the University of California, Berkeley, who studies the issue. Read full story.

Also see  Understanding harmful economic systems., especially the section on obtaining income.

Caption: The offices of Appleby, an offshore law firm, in Hamilton, Bermuda. The company is at the center of leaked documents being called the Paradise Papers. Credit: Meredith Andrews for The New York Times