Models of phenotypic evolution fit to phylogenetic comparative data are widely used to make inferences regarding the tempo and mode of trait evolution. A wide range of models is already available for this type of analysis, and the field is still under active development. One of the most needed developments concerns models that better account for the effect of within- and between-clade interspecific interactions on trait evolution, that can result from processes as diverse as competition, predation, parasitism, or mutualism. Here, we begin by developing a very general comparative phylogenetic framework for (multi)-trait evolution that can be applied to both ultrametric and non-ultrametric trees. This framework not only encapsulates all previous classical models of univariate and multivariate phenotypic evolution, but also paves the way for the consideration of a much broader series of models in which lineages co-evolve, meaning that trait changes in one lineage are influenced by the value of traits in other, interacting lineages. Next, we provide a standard way for deriving the probabilistic distribution of traits at tip branches under our framework. We show that a multivariate normal distribution remains the expected distribution for a broad class of models accounting for interspecific interactions. Our derivations allow us to fit various models efficiently, and in particular greatly reduce the computation time needed to fit the recently proposed phenotype matching model. Finally, we illustrate the utility of our framework by developing a toy model for mutualistic coevolution. Our framework should foster a new era in the study of coevolution from comparative data.