In this thesis the extension of the depth of field of optical systems is investigated. The problem of achieving extended depth of field (EDF) while preserving the transverse resolution is also addressed. A new expression for the transport of intensity equation in the prolate spheroidal coordinates system is derived, with the aim of investigating the phase retrieval problem with applications to EDF. A framework for the optimisation of optical systems with EDF is also introduced, where the main motivation is to find an appropriate scenario that will allow a convex optimisation solution leading to global optima. The relevance in such approach is that it does not depend on the optimisation algorithms since each local optimum is a global one. The multi-objective optimisation framework for optical systems is also discussed, where the main focus is the optimisation of pupil plane masks. The solution for the multi-objective optimisation problem is presented not as a single mask but as a set of masks. Convex frameworks for this problem are further investigated and it is shown that the convex optimisation of pupil plane masks is possible, providing global optima to the optimisation problems for optical systems. Seven masks are provided as examples of the convex optimisation solutions for optical systems, in particular 5 pupil plane masks that achieve EDF by factors of 2, 2.8, 2.9, 4 and 4.3, including two pupil masks that besides of extending the depth of field, are super-resolving in the transverse planes. These are shown as examples of solutions to particular optimisation problems in optical systems, where convexity properties have been given to the original problems to allow a convex optimisation, leading to optimised masks with a global nature in the optimisation scenario.