This is an introductory talk to three different types of cohomologies that play an important role in the theory of complex Kähler manifolds: singular, de Rham and Dolbeault. I will explain how they relate to each other through the framework of sheaf cohomology building a bridge from singular over de Rham to Dolbeault. Then I will talk about the beautiful intricate interplay that arises from combining the integral lattice aspect from singular cohomology with the complex filtration aspect from Dolbeault cohomology under the theme of linearisation. In several examples, I will show how much information can be recovered from this piece of linear algebra (sometimes only conjecturally). Time allowing, I will say something about the algebraic substitutes and analogues of the theory over general characteristics.