We report what is believed to be the first example of fully continuous, 24-hour vertical monitoring of atmospheric optical turbulence. This is achieved using a novel instrument, the 24-hour Shack-Hartmann Image Motion Monitor (24hSHIMM). Optical turbulence is a fundamental limitation for applications such as free-space optical communications, where it limits the achievable bandwidth, and ground-based optical astronomy, restricting the observational precision. Knowledge of the turbulence enables us to select the best sites, design optical instrumentation and optimise the operation of ground-based optical systems. The 24hSHIMM estimates the vertical optical turbulence coherence length, time, angle and Rytov variance from the measurement of a four-layer vertical turbulence profile and a wind speed profile retrieved from meteorological forecasts. To illustrate our advance we show the values of these parameters recorded during a 36-hour, continuous demonstration of the instrument. Due to its portability and ability to work in stronger turbulence, the 24hSHIMM can also operate in urban locations, providing the field with a truly continuous, versatile turbulence monitor for all but the most demanding of applications.