The main objective of this work is to examine in detail basic unsteady pipe flows and to investigate any new physical phenomena. We take the viscoelastic upper-convected Maxwell fluid as our non-Newtonian model and consider the flow of such a fluid in pipes of uniform circular cross-section in the following three cases: (a) when the pressure gradient varies exponentially with time; (b) when the pressure gradient is pulsating; (c) a starting flow under a constant pressure gradient. In the first problem we looked separately at the pressure gradient rising exponentially with time and falling exponentially with time, i.e. the pressure gradient is proportional to e(+/-alpha 2 tau). The behaviour of the flow field depends to a large extent on beta where beta(2) = alpha(2)(1 +/- H alpha(2)) with H being the quotient of the Weissenberg and Reynolds numbers. In both cases for small [beta eta], eta being the radial distance from the axis, the velocity profiles are seen to be parabolic. However, for large \beta eta\ the flows are vastly different. In the case of increasing pressure gradient the flow depicts boundary-layer characteristics while for decreasing pressure gradient the velocity depends on the wall distance. The case of a pulsating pressure gradient is investigated in the second problem. Here the pressure gradient is proportional to cos n tau. Again the flow depends to a large extent on a parameter beta (beta(2) = in - n(2)H). For small values of \beta eta\ the velocity profile is parabolic. However, it is found that, unlike Newtonian fluids, the velocity distribution for the upper-convected Maxwell fluid is not in phase with the exciting pressure distribution. In the case of large \beta eta\ the solution displays a boundary-layer characteristic and the phase of the motion far from the wall is shifted by half a period. The final problem examines a flow that is initially at rest and then set in motion by a constant pressure gradient. A closed form solution has been obtained with the aid of a Fourier-Bessel series. The variation of the velocity across the pipe has been sketched and comparison made with the classical solution.