In the past 50 years, the formalism of L-systems has been successfully used and developed to model the growth of filamentous and branching biological forms. These simulations take place in classical 2-D or 3-D Euclidean spaces. However, various biological forms actually grow in curved, non-Euclidean, spaces. This is for example the case of vein networks growing within curved leaf blades, of unicellular filaments, such as pollen tubes, growing on curved surfaces to fertilize distant ovules, of teeth patterns growing on folded epithelia of animals, of diffusion of chemical or mechanical signals at the surface of plant or animal tissues, etc. In this talk, I will describe how we extended the formalism of L-systems to model the growth of branching structures in curved spaces. We will discuss how the space may feedback on the growing form and contribute to shape. I will also look at examples, where the space in which the form is growing is not necessarily a surface embedded in the euclidean 3-dimensional space, but is rather a space intrinsically curved, i.e. curved but not embedded in any higher-dimensional space. The possibility to use these more abstract Riemannian spaces potentially opens new avenues for formalizing rules driving the morphogenesis of living forms.