Aims. The objective of this study was to examine the kinematics of coronal mass ejections (CMEs) using EUV and coronagraph images, and to make a quantitative comparison with a number of theoretical models. One particular aim was to investigate the acceleration profile of CMEs in the low corona. Methods. We selected two CME events for this study, which occurred on 2006 December 17 (CME06) and 2007 December 31 (CME07). CME06 was observed using the EIT and LASCO instruments on-board SOHO, while CME07 was observed using the SECCHI imaging suite on STEREO. The first step of the analysis was to track the motion of each CME front and derive its velocity and acceleration. We then compared the observational kinematics, along with the information of the associated X-ray emissions from GOES and RHESSI, with the kinematics proposed by three CME models (catastrophe, breakout and toroidal instability). Results. We found that CME06 lasted over eight hours while CME07 released its energy in less than three hours. After the eruption, both CMEs were briefly slowed down before being accelerated again. The peak accelerations during the re-acceleration phase coincided with the peak soft X-ray emissions for both CMEs. Their values were ∼60 m s-2 for CME06 and ∼600 m s-2 for CME07. CME07 reached a maximum speed of over 1000 km s-1 before being slowed down to propagate away at a constant, final speed of ∼700 km s -1. CME06 did not reach a constant speed but was moving at a small acceleration by the end of the observation. Our comparison with the theories suggested that CME06 can be best described by a hybrid of the catastrophe model and breakout model while the characteristics of CME07 were most consistent with the breakout model. Based on the catastrophe model, we deduced that the reconnection rate in the current sheet for CME06 was intermediate, the onset of its eruption occurred at a height of ∼200 Mm, and the Alfvén speed and the magnetic field strength at this height were approximately 130-250 km s-1 and 7 Gauss, respectively.