Event cameras, offering extremely high temporal resolution and high dynamic range, have brought a new perspective to addressing common object detection challenges (e.g., motion blur and low light). However, how to learn a better spatio-temporal representation and exploit rich temporal cues from asynchronous events for object detection still remains an open issue. To address this problem, we propose a novel asynchronous spatio-temporal memory network (ASTMNet) that directly consumes asynchronous events instead of event images prior to processing, which can well detect objects in a continuous manner. Technically, ASTMNet learns an asynchronous attention embedding from the continuous event stream by adopting an adaptive temporal sampling strategy and a temporal attention convolutional module. Besides, a spatio-temporal memory module is designed to exploit rich temporal cues via a lightweight yet efficient inter-weaved recurrent-convolutional architecture. Empirically, it shows that our approach outperforms the state-of-the-art methods using the feed-forward frame-based detectors on three datasets by a large margin (i.e., 7.6% in the KITTI Simulated Dataset, 10.8% in the Gen1 Automotive Dataset, and 10.5% in the 1Mpx Detection Dataset). The results demonstrate that event cameras can perform robust object detection even in cases where conventional cameras fail, e.g., fast motion and challenging light conditions.