The Discontinuity Capturing Petrov-Galerkin (DCPG) / Pressure Stabilised Taylor-Galerkin (PSTG) method for the analysis of flow and heat transfer is presented. The PSTG method circumvents the Babuška-Brezzi restrictions on the interpolation spaces for velocity and pressure, whilst the DCPG method introduces discontinuity capturing (for controlling wiggles) and an implicit subgrid model for LES. The main difference between this work and other stabilised formulations stems from the way the present method is obtained. Some early stabilised methods have been proposed as Petrov-Galerkin formulations with weighting functions defined a priori, whilst more recent formulations are based on the Variational Multiscale (VMS) framework. On the other hand, in the derivation of the DCPG/PSTG method, time discretisations precede the spatial discretisation using finite elements. Furthermore, the method relies on the effective transport velocity, i.e., the projection of the flow velocity on the direction of the gradient of the quantity transported, to introduce a discontinuity capturing effect and an implicit subgrid model suitable for LES. The stabilisation terms that naturally appear in the derivation of the DCPG/PSTG method are proportional to time-steps, which gives insights on how to interpret local time-steps as stabilisation parameters. Numerical examples of flow and heat transfer problems are presented. The examples cover axisymmetric and 2D Cartesian problems in laminar and turbulent flow conditions. Comparison between numerical computations and benchmark results demonstrates the good performance of the DCPG/PSTG method in the simulation of flow and heat transfer problems.