In circular Couette flow between concentric rotating cylinders, two distinct transition types have been identified. The first, called transition by spectral evolution, occurs when the inner cylinder rotates faster than the outer one. As speed increases, secondary modes are excited, starting with Taylor motion (periodic in the axial direction) and then traveling waves (periodic in the circumferential direction). These modes correspond to harmonics of the two fundamental frequencies of the doubly-periodic flow. This transition is a cascade process where energy is transferred through a discrete spectrum to higher frequencies in wave-number space. At high Reynolds numbers, the spectrum becomes continuous. These periodic flows are not uniquely determined by Reynolds number; multiple states (defined by Taylor cells and tangential waves) can exist at a given speed. These states replace each other in a repeatable but irreversible pattern, with vortices and waves appearing or disappearing in pairs. Over 70 transitions have been observed up to 10 times the first critical speed. The angular velocity of tangential waves is nearly constant at 0.34 times the inner cylinder's angular velocity. The second transition, called catastrophic transition, occurs when the outer cylinder rotates faster than the inner one. At fixed Reynolds numbers, the fluid is divided into laminar and turbulent regions separated by interfaces. Turbulent regions may appear and disappear randomly or form regular patterns, such as spiral turbulence rotating at nearly the mean angular velocity of the walls. Spiral turbulence is studied to understand the role of interfaces and intermittency in transition. The paper discusses the historical background of Couette flow, the Rayleigh criterion for instability, and Taylor's work on instability. It describes the experimental investigation of stability and transition in Couette flow, including the design and instrumentation of large and small machines. The large machine uses air as the working fluid and hot-wire anemometry for measurements. The small machine uses silicone oil and suspended-particle flow visualization. The research identified two distinct transition processes: spectral evolution and catastrophic transition. The first is characterized by a gradual transition through a sequence of regular patterns, while the second involves abrupt transitions with laminar-turbulent interfaces. The paper also discusses the tangential wave velocity and the uniqueness of flow states, noting that the boundary between singly- and doubly-periodic flows is not unique but depends on axial wave-number. The study highlights the importance of understanding the mechanisms of transition in Couette flow, including the role of interfaces and intermittency.In circular Couette flow between concentric rotating cylinders, two distinct transition types have been identified. The first, called transition by spectral evolution, occurs when the inner cylinder rotates faster than the outer one. As speed increases, secondary modes are excited, starting with Taylor motion (periodic in the axial direction) and then traveling waves (periodic in the circumferential direction). These modes correspond to harmonics of the two fundamental frequencies of the doubly-periodic flow. This transition is a cascade process where energy is transferred through a discrete spectrum to higher frequencies in wave-number space. At high Reynolds numbers, the spectrum becomes continuous. These periodic flows are not uniquely determined by Reynolds number; multiple states (defined by Taylor cells and tangential waves) can exist at a given speed. These states replace each other in a repeatable but irreversible pattern, with vortices and waves appearing or disappearing in pairs. Over 70 transitions have been observed up to 10 times the first critical speed. The angular velocity of tangential waves is nearly constant at 0.34 times the inner cylinder's angular velocity. The second transition, called catastrophic transition, occurs when the outer cylinder rotates faster than the inner one. At fixed Reynolds numbers, the fluid is divided into laminar and turbulent regions separated by interfaces. Turbulent regions may appear and disappear randomly or form regular patterns, such as spiral turbulence rotating at nearly the mean angular velocity of the walls. Spiral turbulence is studied to understand the role of interfaces and intermittency in transition. The paper discusses the historical background of Couette flow, the Rayleigh criterion for instability, and Taylor's work on instability. It describes the experimental investigation of stability and transition in Couette flow, including the design and instrumentation of large and small machines. The large machine uses air as the working fluid and hot-wire anemometry for measurements. The small machine uses silicone oil and suspended-particle flow visualization. The research identified two distinct transition processes: spectral evolution and catastrophic transition. The first is characterized by a gradual transition through a sequence of regular patterns, while the second involves abrupt transitions with laminar-turbulent interfaces. The paper also discusses the tangential wave velocity and the uniqueness of flow states, noting that the boundary between singly- and doubly-periodic flows is not unique but depends on axial wave-number. The study highlights the importance of understanding the mechanisms of transition in Couette flow, including the role of interfaces and intermittency.