Short answers, pulled from the story.
James Clerk Maxwell published a paper titled On Physical Lines of Force in 1861. This work introduced early concepts of electromagnetic fields and laid the groundwork for what would become known as Maxwell's equations.
Oliver Heaviside transformed Maxwell's original twenty equations into four compact vector calculus forms during the late nineteenth century. He removed unnecessary variables while preserving all physical content of the theory to make the equations rotationally invariant and mathematically transparent.
Wilhelm Eduard Weber and Rudolf Kohlrausch measured electrostatic forces in 1855 experiments using Leyden jars before Maxwell published his full theory. Their calculated velocity matched light speed remarkably closely, confirming Maxwell's predictions about electromagnetic wave propagation.
Microscopic equations express electric and magnetic fields directly in terms of total charge density and current density including atomic-scale contributions. Macroscopic equations introduce auxiliary displacement field D and magnetizing field H to describe large-scale behavior without tracking every atom.
Tensor calculus formulations make Maxwell's equations manifestly compatible with special relativity by treating space and time equally. The electromagnetic tensor F represents components combining electric and magnetic fields into a single antisymmetric covariant order two object within Minkowski spacetime coordinates.