What are the groups that LiAlH4 can and cannot reduce?

Lithium aluminium hydride (LiAIH4) is a nucleophilic reducing agent, and used to reduce polar multiple bonds like >C=O, -CN bond because it is hydride (H-) donor. And it is simply abbreviated as LAH.

LIAIH₄ can reduce:
(1) Aldehydes to primary alcohols,
(2) Ketones to secondary alcohols,
(3) Carboxylic acids and esters to primary alcohols.
(4) Acid chlorides to primary alcohols.
(5) Ester to primary alcohols.
(6) Acid anhydrides to primary alcohols.
(7) Amides and nitriles to primary amines.
(8) Isonitriles to secondary amines.
(9) Epoxides (Cyclic ethers) to alcohols.
(10) lactones (Cyclic esters) to diols. 

Important Notes:
LiAH4 can reduce Aldehydes, ketones, carboxylic acids, ester, acid chlorides, acids anhydrides, acid amides,  Nitro Compounds and nitriles and isonitriles  without attacking (Isolated non polar bonds , like -C=C- ) double bonds present in the Compounds. The only exception in this case alpha- Beta unsaturated compounds containing phenyl group in the Beta position, in this case LiAH₄ also attack on double bond.
We can say that the double or triple bonds in conjugation with the polar multiple bonds can be reduced.