Many blockchain-based protocols, such as Bitcoin, implement a decentralized asset transfer (or exchange) system. As clearly stated in the original paper by Nakamoto, the crux of this problem lies in prohibiting any participant from engaging in double-spending. There seems to be a common belief that consensus is necessary for solving the double-spending problem. Indeed, whether it is for a permissionless or a permissioned environment, the typical solution uses consensus to build a totally ordered ledger of submitted transfers. In this paper we show that this common belief is false: consensus is not needed to implement of a decentralized asset transfer system. We do so by introducing AT2 (Asynchronous Trustworthy Transfers), a class of consensusless algorithms. To show formally that consensus is unnecessary for asset transfers, we consider this problem first in the shared-memory context. We introduce AT2SM, a wait-free algorithm that asynchronously implements asset transfer in the read-write shared-memory model. In other words, we show that the consensus number of an asset-transfer object is one. In the message passing model with Byzantine faults, we introduce a generic asynchronous algorithm called AT2MP and discuss two instantiations of this solution. First, AT2D ensures deterministic guarantees and consequently targets a small scale deployment (tens to hundreds of nodes), typically for a permissioned environment. Second, AT2P provides probabilistic guarantees and scales well to a very large system size (tens of thousands of nodes), ensuring logarithmic latency and communication complexity. Instead of consensus, we construct AT2D and AT2P on top of a broadcast primitive with causal ordering guarantees offering deterministic and probabilistic properties, respectively.