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As the offshore engineering moving from shallow to deep waters, the foundation types for fixed and floating platforms have been gradually evolving to minimize engineering costs and structural risks in the harsh offshore environments. Particular focus of this paper is on the foundation instability and its failure mechanisms as well as the relevant theory advances for the prevail- ing foundation types in both shallow and deep water depths. Piles, spudcans, gravity bases, suction caissons, and plate anchors are detailed in this paper. The failure phenomena and mechanisms for each type of foundations are identified and summarized, respectively. The theoretical approaches along with sophisticated empirical solutions for the bearing capacity problems are then presented. The major challenges are from flow-structure-soil coupling processes, rigorous constitutive modeling of cyclic behaviors of marine sediments, and the spatial variability of soil properties for large-spreading structures. Further researches are suggested to reveal the instability mechanisms for underpinning the evolution of offshore foundations.