  We describe the classical cosmological tests, such as the Log$N$-Log$S$,
redshift-magnitude and angular diameter tests, and propose some new tests of
the evolution of galaxies and the universe. Most analyses of these tests treat
the problem in terms of a luminosity function and its evolution which can lead
to incorrect conclusions when dealing with high redshift sources. We develop a
proper treatment in three parts. In the first part we describe these tests
based on the isophotal values of the quantities such as flux, size or surface
brightness. We show the shortcomings of the simple point source approximation
based solely on the luminosity function and consideration of the flux limit. We
emphasize the multivariate nature of the problem and quantify the effects of
other selection biases due to the surface brightness and angular size
limitations. In these considerations the surface brightness profile plays a
critical role. In the second part we show that considerable simplification over
the complicated isophotal scheme is achieved if these test are carried out in
some sort of metric scheme, for example that suggested by Petrosian (1976).
This scheme, however, is limited to well resolved sources. Finally, we describe
the new tests, which use the data to a fuller extent than the isophotal or
metric based tests, and amount to simply counting the pixels or adding their
intensities as a function of the pixel surface brightness, instead of dealing
with surface brightness, sizes and fluxes of individual galaxies. We show that
the data analysis and its comparison with the theoretical models of the
distributions and evolution of galaxies has the simplicity of the metric test
and utilizes the data more fully than the isophotal test.
