After a brief introduction to local cohomology I am going to discuss the result of my student Yi Zhang on the grading of local cohomology modules in characteristic p>0 and its recent extension to characteristic 0 by Linquan Ma and Wenliang Zhang. Namely, if R is a polynomial ring in n variables over a field and m \subset R is the maximal ideal generated by the variables then it is well-known that H^n_m(R) with its natural grading is isomorphic to E(n), i.e. the naturally graded injective hull E of R/m degree-shifted downward by n. It has also been well-known that if I \subset R is any ideal, then the local cohomology module H^i_m(H^j_I(R)) is isomorphic to a direct sum of a finite number of copies of E. Yi, Linquan and Wenliang sharpened this result by showing that if I \subset R is any homogeneous ideal, then the local cohomology module H^i_m(H^j_I(R)), with its natural grading is isomorphic to a direct sum of a finite number of copies of E(n). Some other related recent results will also be discussed.