[en] In MgB₂ the correlation of microstructure with superconducting properties, in particular the critical current density, requires powerful analytical tools. Critical current densities and electrical resistivities of different MgB₂ superconductors differ by orders of magnitudes and the current limiting mechanisms have not been fully understood. Granularity of MgB₂ is one significant reason for reduced critical current densities and is introduced intrinsically by the anisotropy of B_c2 but also extrinsically by the microstructure of the material. B_c2 enhancement by doping is another important challenge for chemical analysis and, at present, doping levels are not well controlled on the sub-μm scale. In this paper the quantitative electron microscopy and spectroscopy methods essential for the microstructural analysis of MgB₂ are described. By quantitative electron microscopy and spectroscopy we mean a combined SEM and TEM analysis that covers various length scales from μm to nm. Contamination-free sample preparation, chemical mapping including B, and advanced chemical quantification using x-ray microanalysis were essential elements of the applied methodology. The methodology was applied to in situ and ex situ MgB₂ wires and tapes with and without SiC additives. Quantitative B analysis by EDX spectroscopy was applied quantitatively in the SEM and TEM, which is a major achievement. Although MgB₂ is a binary system, the thermodynamics of phase formation is complex, and the complexity is dramatically increased if additives like SiC are used. The small, sub-μm grain sizes of the matrix and secondary phases require TEM methods. However, granularity on the μm scale was also identified and underlines the importance of the combined SEM and TEM studies. Significant differences in the microstructure were observed for in situ and ex situ samples. This holds particularly if SiC was added and yielded Mg₂Si for in situ samples annealed at 600-650 deg. C and Mg-Si-O phases for ex situ samples annealed between 900-1050 deg. C. Only with such a systematic approach combining a large number of microscopy and spectroscopy methods, could a microstructure critical current density model be established that will be presented in another paper. Four microstructural parameters were identified as relevant for the critical current density of wires and tapes and these were: (1) MgB₂ grain size, (2) colony size (a colony is a dense arrangement of MgB₂ grains), (3) oxygen content and (4) volume fraction of B-rich secondary phases. MgB₂ grain size can only be determined by TEM, while colony size, oxygen content and volume fraction of B-rich secondary phases were determined by SEM methods. The formation of oxides was also studied in detail by TEM methods. The importance of electron microscopy methods in the understanding of the thermodynamics of phase formation in MgB₂ as well as in improving the synthesis technology and the superconducting properties of MgB₂ wires and tapes is described. (topical review)