Astronomical observations have played a central role in measuring time and time intervals since antiquity. Modern timekeeping is based on atomic clocks. Although both methods are widely used today, they are not completely compatible with each other. Several solutions have been proposed for reconciling this difference, but none has been accepted to date. I will describe both timekeeping methods and explain the source of the incompatibility. I will conclude with a discussion of why it has been so difficult to find a solution that will satisfy the different user communities.