Logarithmic conformal field theories are based on vertex algebras with non-semisimple representation categories. While examples of such theories have been known for more than 25 years, some crucial aspects of local logarithmic CFTs have been understood only recently, with the help of a description of conformal blocks by modular functors. We present some of these results, both about bulk fields and about boundary fields and boundary states. We also describe some recent progress towards a derived modular functor. This is a summary of work with Terry Gannon, Simon Lentner, Svea Mierach, Gregor Schaumann and Yorck Sommerhäuser.