Grassed waterways are constructed channels covered with vegetation that are used to safely transport excess runoff from fields to water outlets. They are designed based on expected runoff from a 10-year storm event using formulas to calculate discharge and velocity. The design includes determining the shape, size, and slope of the channel to ensure the velocity does not exceed permissible limits based on soil and vegetation type. Construction involves establishing grass cover before use to prevent erosion and maintenance of the grass over time.