In classical auction design, we take it for granted that the auctioneer is trusted and always implements the auction's rules honestly. This assumption, however, no longer holds in modern auctions based on blockchains, or those mediated by third-party platforms such as Google. For example, in blockchain-based auctions, the consensus nodes that partly serve as the auctioneer are incentivized to deviate from honest behavior if profitable. Third-party auction platforms such as Google have also been involved in high-profile anti-trust lawsuits for manipulating their auctions.In this talk, I will describe our recent work on decentralized mechanism design, where we aim to build a new scientific foundation for emerging auctions that are not backed by a trusted auctioneer. I will characterize the mathematical landscape of decentralized mechanism design, by showing several infeasibility and feasibility results. I will also highlight how cryptography can play an essential role for bypassing impossibility results in decentralized mechanism design, leading to a new class of auctions that not only incentivize bidders to act honestly, but also incentivize the auctioneer to play by the book.