In this thesis, two selected topics in magnetism are studied using theoretical modelling and computational methods. The first of these is the magnetocrystalline anisotropy energy (MAE) of transition metal based magnets. In particular, ways of finding 3d transition metal based materials with large MAE are considered. This is motivated by the need for new permanent magnet materials, not containing rare-earth elements, but is also of interest for other technological applications, where the MAE is a key quantity. The mechanisms of the MAE in the relevant materials are reviewed and approaches to increasing this quantity are discussed. Computational methods, largely based on density functional theory (DFT), are applied to guide the search for relevant materials. The computational work suggests that the MAE of Fe1-xCox alloys can be significantly enhanced by introducing a tetragonality with interstitial B or C impurities. This is also experimentally corroborated. Alloying is considered as a method of tuning the electronic structure around the Fermi energy and thus also the MAE, for example in the tetragonal compound (Fe1-xCox)2B. Additionally, it is shown that small amounts (2.5-5 at.%) of various 5d dopants on the Fe/Co-site can enhance the MAE of this material with as much as 70%. The magnetic properties of several technologically interesting, chemically ordered, L10 structured binary compounds, tetragonal Fe5Si1-xPxB2 and Hexagonal Laves phase Fe2Ta1-xWx are also investigated. The second topic studied is that of magnetic effects on the elastic scattering of fast electrons, in the context of transmission electron microscopy (TEM). A multislice solution is implemented for a paraxial version of the Pauli equation. Simulations require the magnetic fields in the sample as input. A realistic description of magnetism in a solid, for this purpose, is derived in a scheme starting from a DFT calculation of the spin density or density matrix. Calculations are performed for electron vortex beams passing through magnetic solids and a magnetic signal, defined as a difference in intensity for opposite orbital angular momentum beams, integrated over a disk in the diffraction plane, is observed. For nanometer sized electron vortex beams carrying orbital angular momentum of a few tens of ħ, a relative magnetic signal of order 10-3 is found. This is considered realistic to be observed in experiments. In addition to electron vortex beams, spin polarised and phase aberrated electron beams are considered and also for these a magnetic signal, albeit weaker than that of the vortex beams, can be obtained. / <p>Felaktigt ISBN i den tryckta versionen: 9789155497149</p><p></p>