The modern world is heavily reliant on electromagnetic devices to convert mechanical energy into electrical energy and vice versa. These devices are fundamental to powering our society, and the growing need for automated production lines and electrified transportation is driving the demand for even more advanced electromagnetic actuators. A critical component of these devices is the electromagnetic coil, typically constructed from copper wire wound into simple shapes. Over the years, extensive research and industrial development have led to a deep understanding of this technology. Nevertheless, innovation is needed to improve upon state-of-the-art or to access applications with harsher requirements. In this thesis, we propose using additive manufacturing techniques to fabricate electromagnetic coils, which eliminates geometric limitations and opens up new design possibilities. We also utilize computational design methods, which reformulate the design problem as an optimization one. These tools generate complex shapes belonging to a broader solution set than the one accessible by a human designer. The implemented method (topology optimization) investigates the ideal distribution of material in space. It first demonstrates the possibility of generating coils within a 2-D plane. This framework is expanded to consider the packing of multiple coils constituting motor windings without a-priori knowledge of their position or shape. A 2.5-D approach is formulated to design overlapping coils while maintaining a low computational cost. A 3-D analysis is also explored for designing coils with different functions in an electromagnet manipulator. The generated designs exhibit significantly improved electromagnetic performance, such as a motor constant increased by 17% or regions with uniform magnetic field 7 times larger than the references. The feasibility of additively manufacturing electromagnetic coils is examined through a study case of a linear motor with a reference coil topology. Innovative design features, such as tracks with an evolving cross-section and "completed" linear windings, are proposed to fully harvest the design freedom provided by additive manufacturing. Additionally, we propose the concept of multi-functional windings, integrating heat sinks within the electromagnetic coils for improved efficiency and self-cooling capabilities. This results in the possibility of increasing the motor force by 17.3% without increasing the volume of permanent magnets. Prototypes of various coil geometries are fabricated and tested. Overall, this thesis lays the foundation for a new generation of electromagnetic devices with improved performance through the combination of computational design methods and additive manufacturing technologies.