True homeopathic research is done through provings.
These establish the therapeutic efficacy of the medicine being tested.
New provings are being done at an increasing rate. For some examples of
high quality provings, visit the website of the Dynamis
School
.

Research into cured clinical cases corroborates the provings
data, and expands on them.

Research as it is more traditionally understood in mainstream
medicine is also being done. The National
Center for Homeopathy
has a website which provides an introduction
into such research.  There is a lot of such research that has been done in the last several years. 

I have also posted scanned images of a compilation of current research in homeopathy, compiled by Iris Bell, M.D., of the University of Arizona Program in Integrative Medicine.  These can be viewed here.  The slide show Dr. Bell presented can be viewed (as a pdf file) here.