Context. We present the discovery of two transiting extrasolar planets by the satellite CoRoT. Aims: We aim at a characterization of the planetary bulk parameters, which allow us to further investigate the formation and evolution of the planetary systems and the main properties of the host stars. Methods: We used the transit light curve to characterize the planetary parameters relative to the stellar parameters. The analysis of HARPS spectra established the planetary nature of the detections, providing their masses. Further photometric and spectroscopic ground-based observations provided stellar parameters (log g, Teff, v sin i) to characterize the host stars. Our model takes the geometry of the transit to constrain the stellar density into account, which when linked to stellar evolutionary models, determines the bulk parameters of the star. Because of the asymmetric shape of the light curve of one of the planets, we had to include the possibility in our model that the stellar surface was not strictly spherical. Results: We present the planetary parameters of CoRoT-28b, a Jupiter-sized planet (mass 0.484 ± 0.087 MJup; radius 0.955 ± 0.066 RJup) orbiting an evolved star with an orbital period of 5.208 51 ± 0.000 38 days, and CoRoT-29b, another Jupiter-sized planet (mass 0.85 ± 0.20 MJup; radius 0.90 ± 0.16 RJup) orbiting an oblate star with an orbital period of 2.850 570 ± 0.000 006 days. The reason behind the asymmetry of the transit shape is not understood at this point. Conclusions: These two new planetary systems have very interesting properties and deserve further study, particularly in the case of the star CoRoT-29. The CoRoT space mission, launched on December 27th 2006, was developed and is operated by CNES, with the contribution of Austria, Belgium, Brazil, ESA (RSSD and Science Programme), Germany, and Spain. Based on observations obtained with the Nordic Optical Telescope, operated on the island of La Palma jointly by Denmark, Finland, Iceland, Norway, and Sweden, in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias, in time allocated by OPTICON and the Spanish Time Allocation Committee (CAT). The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement number RG226604 (OPTICON). This work makes use of observations from the LCOGT network.Appendices are available in electronic form at http://www.aanda.org