Originating from geometry and topology, the so-called method of convex integration has been recognised as a powerful tool for the study of nonlinear PDEs and variational problems. Generally speaking, it is used to obtain an abundance of exact solutions of certain problems by means of an iteration process, starting from a suitable relaxation of the problem. I will discuss the method and two of its recent applications: On the one hand, C. De Lellis and L. Székelyhidi used convex integration methods to construct solutions of the incompressible Euler equations with surprising properties. On the other hand, together with K. Koumatos and F. Rindler I obtained various results on the flexibility of the pointwise Jacobian determinant in subcritical Sobolev spaces. We will see how these seemingly unrelated issues can both be tackled by convex integration strategies.