Abstract: Emerging quantum networks enable distributed quantum computing (DQC) by interconnecting modest, noisy quantum processors. However, practical deployments face tight memory budgets, probabilistic entanglement generation, and decoherence, all while serving multiple users. In realistic settings -- cloud execution, iterative/interactive workloads and concurrent jobs -- circuits do not arrive in a single batch but as a stream. This creates a fundamentally online problem: the system must react to stochastic arrivals, balance latency and throughput, and remain stable under varying loads without foresight of future jobs. We study Online Scheduling and Distribution of Quantum Circuits (OS-DQC) and formalize the online scheduling problem for quantum networks: circuits arrive over time and must be executed without knowledge of future arrivals. At each event time (arrival or completion), the system decides which waiting circuits to launch and how to place their qubits, while accounting for congestion from in-flight circuits and constraints due to finite memories and decoherence. The objective is to minimize the mean completion time and makespan under these physical and resource constraints.