The authors study a class of centered Gaussian processes on \([0,\infty)\) which they call ``sub-fractional Brownian motions''. The covariance function is given by \[ s^h + t^h -\tfrac{1}{2}\left[ (s+t)^h +| s-t| ^h\right]\;,\quad t,s\geq 0\;, \] for a certain \(h\in (0,2)\). Of course, if \(h=1\), one gets the ordinary Brownian motion. Properties of those processes are stated and proved, for example, self-similarity and path properties. Finally, it is shown how those processes arise in the investigation of occupation time fluctuations of some certain particle systems.