Electromagnetic compatibility remains an important topic in the design and manufacturing of printed circuit boards (PCBs). Compatibility of these devices with their surroundings is becoming increasingly difficult as a modern PCB can have hundreds or thousands of parts, operating on many layers, and all at high speed. One such part is a via and its clearance, or via hole, commonly required in multilayer circuits where vertical connections between layers are used. The via hole may be exposed to large electromagnetic fields within the PCB. Although electrically small, the via hole provides a pathway for the fields to excite the exterior, either directly or through coupling to adjacent structures. To quantify this process, the near fields and radiation of an excited via hole are analysed, and are the focus of this thesis. The near fields of the via hole are first decoupled into electric and magnetic fields of the 'static' type. In both cases a series solution for two regions, one outside, and one inside the layers is constructed. The coefficients of the terms of the series are chosen to best satisfy the boundary behaviour of the fields on the conducting surfaces and across the hole. The criteria for assessing quality of the solution is based on the least squares method (LSM). Linear equation systems for both models are derived, and as no numerical integration or discretisation is required, an efficient and robust implementation to find the near fields is developed. Transformation into the far field is then achieved through surface integration of relevant field quantities close to the via hole. The far fields are best viewed as that due to two dipoles, of the magnetic and electric type, with strength and orientation depending on how the via hole is excited. It is shown that the two dipole model is sufficient to find the radiation from a 1mm diameter via hole at a frequency up to 8 GHz. Of further interest is how the choice of via hole dimensions affects the dipole moments and the near fields solved earlier are a key to this understanding.