The evolution of the stars in globular clusters has been followed from the main sequence to the top Qf the red giant branch. In the initial phases the relevant models consist of partially degenerate isothermal helium cores and radiative hydrogen envelopes. These models show the well-known turnoff from tile main sequence in the Hertasprung-Russell diagram of globular clusters. The helium core steadily increases in mass because of the hydrogen-burning. When the core has reached approximately 20 per cent of the total stellar mass, tile envelope has reached an appreciable extent, and it is found necessary to take into account explicitly the photospheric boundary condition and the hydrogen convection zone. During the subsequent phases this convection zone steadily deepens while the luminosity increases; the star evolves in the Hertxsprung-Russell diagram along the observed red giant sequence. During these phases the internal temperature steadily rises so that the carbon cycle can keep in balance with the increasing luminosity. At the top of the red giant sequence the temperature in the hydrogen-burning shell at the edge of the core reaches about 40,000,000 , while the helium core reaches approximately 50 per cent of the stellar mass. The model sequence thus computed appears to be' in satisfactory agreement with the observations in globular clusters. In addition, the difference between the red giants of population I and \ II can, it seems, be explained by the difference in the metal abundance of the two populations if this abundance difference is taken account of in the photospheric boundary condition; for the relevant evolution phases, a larger metal content produces a larger extent of the envelope and hence later spectral types. It is estimated that near the top of the red giant sequence the more and more rapid contraction of the core will heat the center until it reaches the temperature necessary to start helium-burning. The onset of this process is believed to terminate the red giant sequence by eliminating the degeneracy in the core. Preliminary computations suggest that this change in the stellar model will rather suddenly lower the luminosity and put the star on the beginning of the horizontal branch in the Hertxsprung-Russell diagram.