By providing replacements for damaged tissues and organs, in vitro tissue engineering has the potential to become a viable alternative to donor-provided organ transplant, which is increasingly hampered by a shortage of available tissue. The complexity of the myriad biophysical and biochemical processes that together regulate tissue growth renders almost impossible understanding by experimental investigation alone. Mathematical modelling applied to tissue engineering represents a powerful tool with which to investigate how the different underlying processes interact to produce functional tissues for implantation. The aim of this review is to demonstrate how a combination of mathematical modelling, analysis and in silico computation, undertaken in collaboration with experimental studies, may lead to significant advances in our understanding of the fundamental processes that regulate biological tissue growth and the optimal design of in vitro methods for generating replacement tissues that are fully functional. With this in mind, we review the state-of-the-art in theoretical research in the field of in vitro tissue engineering, concentrating on continuum modelling of cell culture in bioreactor systems and with particular emphasis on the generation of new tissues from cells seeded on porous scaffolds. We highlight the advantages and limitations of different mathematical modelling approaches that can be used to study aspects of cell population growth. We also discuss future mathematical and computational challenges and interesting open questions.