Heat and mass transfer in fluid flows traditionally is examined in terms of temperature and concentration fields and heat/mass-transfer coefficients at fluid-solid interfaces. However, heat/mass transfer may alternatively be considered as the transport of a passive scalar by the total advective-diffusive flux in a way analogous to the transport of fluid by the flow field. This Lagrangian approach facilitates heat/mass-transfer visualisation in a similar manner as flow visualisation and has great potential for transport problems in which insight into (interaction between) the scalar fluxes throughout the entire configuration is essential. This ansatz furthermore admits investigation of heat and mass transfer by well-established geometrical methods from laminar-mixing studies, which offers promising new research capabilities. The Lagrangian approach is introduced and demonstrated by way of representative examples.