A popular line of research in evolutionary biology is to use time-calibrated phylogenies to understand the process of diversification. This is done by performing statistical inference under stochastic models of species diversification. These models thus need to be robust, biologically sound and mathematically tractable. We first introduce some new lineage-based, stochastic models of phylogenies, featuring e.g., protracted speciation or age-dependent extinction. Using recent mathematical results allowing the computation of tree likelihoods, we present ML parameter estimates inferred from the bird phylogeny. Our goal is then to obtain (these or other) models of phylogenies, starting from an individual-based description of populations. We present in particular two non-exchangeable models of phylogenetic trees thus obtained. In the first one, speciation is modelled by genetic differentiation of individual lineages. The second one is a scaling limit of the Tilman-Levins type model where interspecific competition is only felt by older species from younger species.