Many of the challenges that limited aero-engine operation in the 1950s, 60s, 70s and 80s were static in nature: hot components exceeding temperature margins, stresses in the high-speed rotating structure approaching safety limits, and turbomachinery aerodynamic efficiencies missing performance goals. Modeling tools have greatly improved since and have helped enhance jet engine design, largely due to better computers and improved simulations of the fluid flow and supporting structure. The situation is thus different today, where important problems encountered past the design and development phases are dynamic in nature. These can jeopardize engine certification and lead to major delays and increased program cost. A real challenge is the characterization of damping and the related dynamic behavior of rotating and stationary components and assemblies, and of the fluid-structure interactions and coupling. The theme of this lecture is instability in the broadest sense. A number of problems of technological interest in aero-engines are discussed with focus on dynamical system modeling and identification of the underlying mechanisms. Future perspectives on outstanding seminal problems and grand challenges are also given.