We present the analysis of the radial gradients of stellar absorption lines in a sample of 32 bulges of edge-on spiral galaxies, spanning nearly the full Hubble sequence (from S0 to Sc types), and a large range of velocity dispersion (from about 60 to 300km/s). Different diagnostics such as index-index, gradient-gradient diagrams, and simple stellar population models are used to tackle the origin of the variation of the bulge stellar population. We find that the vast majority of bulges show older age, lower metallicity and higher [alpha/Fe] in their outer regions than in their central parts. The radial gradients in [Fe/H] are 2 to 3 times larger than in Log(age). The relation between gradient and bulge velocity dispersion is interpreted as a gradual build up of the gradient mean values and their dispersions from high to low velocity dispersion, rather than a pure correlation. The bulge effective radii and the Hubble type of the parent galaxies seem to play a more minor role in causing the observed spatial distributions. At a given velocity dispersion, bulges and ellipticals share common properties.