Motivated by the proliferation of real-time applications in multimedia communication systems, tactile Internet, and cyber-physical systems, supporting delay-constrained traffic becomes critical for such systems. In delay-constrained traffic, each packet has a hard deadline; when it is not delivered before its deadline is up, it becomes useless and will be removed from the system. In this work, we focus on designing random access schemes for delay-constrained wireless communications. We first investigate three ALOHA-based schemes and prove that the system timely throughput of all three schemes under corresponding optimal transmission probabilities asymptotically converges to $1/e$, same as the well-known throughput limit for delay-unconstrained ALOHA systems. The fundamental reason why ALOHA-based schemes cannot achieve asymptotical system timely throughput beyond $1/e$ is that all active ALOHA stations access the channel with the same probability in any slot. To go beyond $1/e$, we propose a reinforcement-learning-based scheme for delay-constrained wireless communications, called RLRA-DC, under which different stations collaboratively attain different transmission probabilities by only interacting with the access point. Our numerical result shows that the system timely throughput of RLRA-DC can be as high as 0.8 for tens of stations and can still reach 0.6 even for thousands of stations, much larger than $1/e$.