Plants have evolved specialized vascular tissues for the distribution of energy, water, nutrients, and for communication. The phloem transports sugars from photosynthetic source regions (e.g. mature leaves) to sugar sinks (e.g. developing tissues such as buds, flowers, roots). Moreover, chemical signals such as hormones, RNAs and proteins also move in the phloem. Basic physical processes strongly limit phloem anatomy and function. This paper provides an overview of recent research and perspectives on phloem biomechanics and the physical constraints relevant to sugar transport in plants.