Abstract: We present almost-surely terminating asynchronous Byzantine agreement (ABA) protocols with a constant expected running time, in two different settings. The first protocol is in a completely asynchronous setting and has non-optimal resilience, outperforming the communication complexity of all the existing protocols in the asynchronous setting. The second protocol has optimal resilience and considers a hybrid setting for the first time, where the parties are available with one synchronous round at the beginning of the protocol execution, after which the network is completely asynchronous.