In addition to ever-present thermal noise, communication and sensor systems can contain significant amounts of interference with outlier (e.g. impulsive) characteristics. Such outlier interference (including that caused by nonlinear signal distortions, e.g. clipping) can be efficiently mitigated in real-time using intermittently nonlinear filters. Depending on the interference nature and composition, improvements in the quality of the signal of interest achieved by such filtering will vary from "no harm" to substantial. In this tutorial, we explain in detail why the underlying outlier nature of interference often remains obscured, discussing the many challenges and misconceptions associated with state-of-art analog and/or digital nonlinear mitigation techniques, especially when addressing complex practical interference scenarios. We then focus on the methodology and tools for real-time outlier noise mitigation, demonstrating how the "excess band" observation of outlier noise enables its efficient in-band mitigation. We introduce the basic real-time nonlinear components that are used for outlier noise filtering and provide examples of their implementation. We further describe complementary nonlinear filtering arrangements for wide- and narrow-band outlier noise reduction, providing illustrations of their performance and the effect on channel capacity. Finally, we outline "effectively analog" digital implementations of these filtering structures, discuss their broader applications, and comment on the ongoing development of the platform for their demonstration and testing. To emphasize the effectiveness and versatility of this approach, in our examples we use particularly challenging waveforms that severely obscure low-amplitude outlier noise, such as broadband chirp signals (e.g. used in radar, sonar, and spread-spectrum communications) and "bursty," high crest factor signals (e.g. OFDM).