The aim of this research is to study free transverse vibrations of an elliptic solid thin plate with a varying eccentricity and an elliptic plate with an elliptic cutout with the same eccentricity. Both analytical and numerical methods are used to examine the problem. For plates with small eccentricity and/or small central elliptic cutout asymptotic methods are used to describe the effect of the eccentricity and cutout on the vibrational spectra. For a circular plate a vibrational mode can be expressed by one particular solution of the differential equation governing vibrations, for an elliptic plate all appropriate particular solutions should be summed up to express even one vibrational mode. That makes the problem much more difficult. The main aim of the research is to follow what happens to the spectrum of transverse vibration frequencies of a solid circular plate with increasing of the central hole area and/or the plate eccentricity. Frequencies and modes were also found by finite element method using the package ANSYS 18.1. The results were compared with those obtained by other authors who used Rayleigh-Ritz and other numerical and analytical methods.