Morphogenesis, the formation of biological shape and pattern during embryonic development, is a topic of intensive experimental investigation, so the participating cell types and molecular signals continue to be characterized in great detail. Yet this only partly tells biologists how molecules and cells interact dynamically to construct a biological tissue. Mathematical and computational modeling are a great help in answering such questions on biological morphogenesis. Cell-based simulation models of blood vessel growth describe the behavior of cells and the signals they produce. They then simulate the collective behavior emerging from these cell-cell interactions. In this way cell-based models help analyze how cells assemble into biological structures, and reveal the microenvironment the cells produce collectively feeds back on individual cell behavior. In this way, our simulation models, based on a Cellular Potts model combined with partial-differential equations, have shown that the elongated shape of cells is key to correct spatiotemporal in silico replication of vascular network growth. The models have also helped identify a new stochastic mechanism for the formation of branched structures in epithelial gland tissues. I will discuss some recent insights into these mechanisms. Then I will discuss our more recent cell-based modeling studies of cell-extracellular matrix interactions during angiogenesis. I will conclude by suggesting some interesting continuum and stochastic mathematical problems that our cell-based simulations suggest.