Chapters 1-4 are the lecture notes of my course "Real Algebraic Geometry I" from the winter term 2020/2021. Chapters 5-8 are the lecture notes of its continuation "Real Algebraic Geometry II" from the summer term 2021. Chapters 9-10 are the lecture notes of its further continuation "Geometry of Linear Matrix Inequalities" from the winter term 2021/2022.Because of the COVID-19 pandemic, I produced a series of accompanying screencasts of varying length where go in detail through the material. These can be found within a playlist of my Youtube Channel https://www.youtube.com/channel/UCcsp7yAQjJQHx3SN3aL_r1AOther material like the exercise sheets are available somewhere on my website: