Integrated sensing and communication (ISAC) can substantially improve spectral, hardware, and energy efficiency by unifying radar sensing and data communications. In wideband and scattering-rich environments, clutter often dominates weak target reflections and becomes a fundamental bottleneck for reliable sensing. Practical ISAC clutter includes "cold" clutter arising from environmental backscatter of the probing waveform, and "hot" clutter induced by external interference and reflections from the environment whose statistics can vary rapidly over time. In this article, we develop a unified wideband multiple-input multiple-output orthogonal frequency-division multiplexing (MIMO-OFDM) signal model that captures both clutter types across the space, time, and frequency domains. Building on this model, we review clutter characterization at multiple levels, including amplitude statistics, robust spherically invariant random vector (SIRV) modeling, and structured covariance representations suitable for limited-snapshot regimes. We then summarize receiver-side suppression methods in the temporal and spatial domains, together with extensions to space-time adaptive processing (STAP) and space-frequency-time adaptive processing (SFTAP), and we provide guidance on selecting techniques under different waveform and interference conditions. To move beyond reactive suppression, we discuss clutter-aware transceiver co-design that couples beamforming and waveform optimization with practical communication quality-of-service (QoS) constraints to enable proactive clutter avoidance. We conclude with open challenges and research directions toward environment-adaptive and clutter-resilient ISAC for next-generation networks.