A fundamental question in modern cell biology is how cellular and subcellular structures are formed and maintained given their particular molecular components. How are the different shapes, sizes, and functions of cellular organelles determined, and why are specific structures formed at particular locations and stages of the life cycle of a cell? In order to address these questions, it is necessary to consider the theory of self-organizing non-equilibrium systems. We are particularly interested in identifying and analyzing novel mechanisms for pattern formation that go beyond the standard Turing mechanism and diffusion-based mechanisms of protein gradient formation. In this talk we present three examples of non-classical biological pattern formation: (i) Space-dependent switching diffusivities and cytoplasmic protein gradients in the C. elegans zygote (ii) Transport models of cytoneme-based morphogenesis. (iii) Hybrid Turing mechanism for the homeostatic control of synaptogenesis in C. elegans.