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With increasing global energy consumption, stringent environmental protection legislation and safety regulations in industrialised nations, energy saving has been put under high priority. One of the most efficient ways of energy reduction is through heat transfer enhancement for additional heat recovery. Applying compact heat exchanger is one of the main strategies of heat transfer enhancement. However, the application of compact heat exchangers is prohibited by the lack of design methodology. Therefore, the aim of this research is to tackle the problem of developing optimisation methodologies of plate/plate-fin heat exchanger design. A mathematical model of plate-fin heat exchanger design is proposed to consider fin type selection with detailed geometry and imposed constraints simultaneously. The concept of mix-and-match fin type combinations is put forward to include all possible fin type combinations in a heat exchanger. The mixed integer nonlinear programming (MINLP) model can be converted to a nonlinear programming (NLP) model by employing continuous heat transfer and pressure drop correlations and considering the basic fin geometric parameters as continuous variables. The whole optimisation is based on volumetric minimisation or capital cost minimisation and completed by CONOPT solver in GAMS. Case studies are carried out to demonstrate the effectiveness and benefits of the new proposed methodology. For plate heat exchangers, the design methodology is developed on the basis of plate-fin heat exchanger methodology, and takes phase change, plate pattern selection, flow arrangement and pressure drop constraints simultaneously. The phase change problem is tackled by dividing the whole process into several subsections and considering constant physical properties in each subsection. The performances of various flow arrangements are evaluated by correction factors of logarithmic mean temperature difference. For two-phase conditions, the heat transfer and pressure drop performance are predicted by continuous two-phase Nusselt number and Fanning friction factor correlations to avoid the MINLP problem. The optimisation is solved by CONOPT solver as well. The feasibility and accuracy of the new proposed methodology is examined by case studies.