Towards the end of their lives, stars like the Sun greatly expand to become red giant stars. Such evolved stars could provide stringent tests of stellar theory, as many uncertainties of the internal stellar structure accumulate with age. Important examples are convective overshooting and rotational mixing during the central hydrogen-burning phase, which determine the mass of the helium core, but which are not well understood(1). In principle, analysis of radial and non-radial stellar oscillations can be used to constrain the mass of the helium core. Although all giants are expected to oscillate(2), it has hitherto been unclear whether non-radial modes are observable at all in red giants, or whether the oscillation modes have a short or a long mode lifetime(3-7), which determines the observational precision of the frequencies. Here we report the presence of radial and non-radial oscillations in more than 300 giant stars. For at least some of the giants, the mode lifetimes are of the order of a month. We observe giant stars with equally spaced frequency peaks in the Fourier spectrum of the time series, as well as giants for which the spectrum seems to be more complex. No satisfactory theoretical explanation currently exists for our observations.