We show that in a source-free subwavelength region of microwave fields, there can exist field structures with a local coupling between the time-varying electric and magnetic fields differing from the electric-magnetic coupling in regular-propagating free-space electromagnetic waves. To distinguish such field structures from regular electromagnetic (EM) field structures, we term them as magnetoelectric (ME) fields. We study a structure and conservation laws of microwave ME near fields. We show that there exist sources of microwave ME near fields - the ME particles. These particles are represented by small quasi-two-dimensional ferrite disks with magnetic-dipolar-oscillation spectra. The near fields originating from such particles are characterized by topologically distinctive power-flow vortices, nonzero helicity, and a torsion degree of freedom. The paper consists of two main parts. In the first one, we give a theoretical background of properties of the electric and magnetic fields inside and outside of a ferrite particle with magnetic-dipolar-oscillation spectra resulting in the appearance of microwave ME near fields. In the second main part, we represent numerical and experimental studies of the microwave ME near fields and their interactions with matter. Based on the obtained properties of the ME near fields, we discuss possibilities for effective microwave sensing of natural and artificial chiral structures.