dotty is a new, experimental Scala compiler based on DOT, the calculus of Dependent Object Types. Higher-kinded types are a natural extension of first-order lambda calculus, and have been a core construct of Haskell and Scala. As long as such types are just partial applications of generic classes, they can be given a meaning in DOT relatively straightforwardly. But general lambdas on the type level require extensions of the DOT calculus to be expressible. This paper is an experience report where we describe and discuss four implementation strategies that we have tried out in the last three years. Each strategy was fully implemented in the dotty compiler. We discuss the usability and expressive power of each scheme, and give some indications about the amount of implementation difficulties encountered.