Author’s abstract: Consider a Markov process \(X\) on \([0,\infty )\) which has only negative jumps and converges as time tends to infinity a.s. We interpret \(X(t)\) as the size of a typical cell at time t, and each jump as a birth event. More precisely, if \(\Delta X(s)=-y<0\), then \(s\) is the birthtime of a daughter cell with size \(y\) which then evolves independently and according to the same dynamics, that is, giving birth in turn to great-daughters, and so on. After having constructed rigorously such cell systems as a general branching process, we define growth-fragmentation processes by considering the family of sizes of cells alive a some fixed time. We introduce the notion of excessive functions for the latter, whose existence provides a natural sufficient condition for the non-explosion of the system. We establish a simple criterion for excessiveness in terms of \(X\). The case when \(X\) is self-similar is treated in details, and connexions with self-similar fragmentations and compensated fragmentations are emphasized.