Summary: Runtime Enforcement (RE) is a technique aimed at monitoring the executions of a system at runtime and ensure its compliance against a set of formal requirements (properties). RE employs an enforcer (a safety wrapper for the system) which modifies the (untrustworthy) output by performing actions such as delaying (by storing/buffering) and suppressing events, when needed. In this paper, to handle practical applications with memory constraints, we propose a new RE paradigm where the memory of the enforcer is bounded/finite. Besides the property to be enforced, the user specifies a bound on the enforcer memory. Bounding the memory poses various challenges such as how to handle the situation when the memory is full, how to optimally discard events from the buffer to accommodate new events and let the enforcer continue operating. We define the bounded-memory RE problem and develop a framework for any regular property. All of our results are formalized and proved. We also analyze probabilistically how much memory is required on an average case for a given regular property, such that the output of the bounded enforcer is equal to that of the unbounded enforcer up to a fixed probability. The proposed framework is implemented and a case study is worked out to show the practicability and usefulness of the bounded enforcer in the real-world and to show the usage of the aforementioned probabilistic analysis on them. The performance is evaluated via some examples from application scenarios and it indicates linear changes in the execution time of the enforcers in response to increases in trace length, property complexity, and buffer sizes.