It is well established that a small number of firms, known as fast-growth firms or Gazelles, create most new jobs. Despite the importance of this topic from a policy point of view, most studies are descriptive and explore a limited number of characteristics of fast-growth firms. The existence of some correlation between two or more of the determinants of fast growth could yield, however, spurious results. To avoid that problem, this paper performs a multivariate analysis of the determinants of fast growth using a panel of Spanish firms. The variables explored include sector of activity, region and newness of the firm as well as access to external finance and firms' human resource practices. We control for the presence of unobserved time-invariant, firm-specific heterogeneity as well as for the possible existence of state dependence. We find that past extreme growth episodes increase the probability of current fast growth, which is in contrast to previous findings on the topic. We also find that human resource practices, such as employing qualified personnel or the mix of contracts offered, are important determinants of fast growth. Lastly, newness and access to credit are found to be important to explain firm growth, but they are not significant determinants of fast or extreme employment growth, thereby reflecting the existence of non-linearities in the growth process of firms.