We study the primordial gravitational wave background produced in models of single field inflation. Using the inflationary flow approach, we investigate the amplitude of gravitational wave spectrum in the frequency range 1 mHz - 1 Hz pertinent to future space-based laser interferometers. For models that satisfy the current observational constraint on the tensor-to-scalar ratio, r<0.36, we derive a strict upper bound of omega_{gw}<1.6 x 10^{-15}, independent of the form of the inflationary potential. Applying, in addition, the observational constraints on the spectral index and its running, omega_{gw} is expected to be considerably lower than this bound unless the shape of the potential is finely tuned. We contrast our numerical results with those based on simple power-law extrapolation of the tensor power spectrum from CMB scales. In addition to single field inflation, we summarise a number of other possible cosmological sources of primordial gravitational waves and assess what might be learnt from direct detection experiments such as LISA, Big Bang Observer and beyond.