For Kalman filters to be applied correctly, it is important to know the boundary conditions under which the Kalman equations may be used. This means that the specific problem to be solved must be checked accordingly. If these prerequisites are not met, the Kalman equations will not yield the desired result. In this chapter, the fundamental equations of the Kalman filter are derived, and at each relevant point, the necessary prerequisites are highlighted. The chapter begins by describing the structure of a classical Kalman filter, which is based on the state-space representation of a real system, including measurement and system noise. Subsequently, the fundamental equations, divided into prediction and correction, are derived. Finally, an alternative calculation of the Kalman gain is presented. This is particularly important as a fundamental equation for the adaptive Kalman filter (ROSE filter).