  In this paper we give a survey of results on various analytical aspects of
time fractional diffusion equations. We describe the approach via abstract
Volterra equations and collect results on strong solutions in the $L_p$ sense.
We further discuss the concept of weak solutions for equations with rough
coefficients and give an account of recent developments towards a De
Giorgi-Nash-Moser theory for such equations. The last part summarizes recent
results on the long-time behaviour of solutions, which turns out to be
significantly different from that in the heat equation case.
