Developing a theory of bisimulation in higher-order languages can be
hard. Particularly challenging can be the proof of congruence and,
related to this, enhancements of the bisimulation proof method with
``up-to context'' techniques.
 
We present _logical bisimulations_, a form of bisimulation for
higher-order languages, in which the bisimulation clause is somehow
reminiscent of logical relations.  We consider purely functional
languages, in particular untyped call-by-name and call-by-value
lambda-calculi, and, in each case: we present the basic properties of
logical bisimilarity, including congruence; we show that it coincides
with contextual equivalence; we develop some up-to techniques,
including up-to context, as examples of possible enhancements of the
associated bisimulation method.