The formation and growth of multi-cellular organisms and tissues from several genetically identical embryo cells is one of the most fundamental natural phenomena. These processes are stimulated and governed by multiple biological signaling molecules, which are also called morphogens. Embryo cells are able to read and pass genetic information by measuring the non-uniform concentration profiles of signaling molecules. It is widely believed that the establishment of concentration profiles of morphogens, commonly referred as morphogen gradients, is a result of complex biophysical and biochemical processes that might involve diffusion and degradation of locally produced signaling molecules. In this review, we discuss various theoretical aspects of the mechanisms for morphogen gradient formation, including stationary and transient dynamics, the effect of source delocalization, diffusion, different degradation mechanisms, and the role of spatial dimensions. Theoretical predictions are compared with experimental observations. In addition, we analyze the potential alternative mechanisms of the delivery of biological signals in embryo cells and tissues. Current challenges in understanding the mechanisms of morphogen gradients and future directions are also discussed.