Additive number theory is currently an active field related to combinatorics. In this paper we give a survey of problems and results concerning lower bounds for cardinalities of various restricted sumsets with elements in a field or an abelian group. 1. Erdős-Heilbronn conjecture and the polynomial method Let A = {a1, . . . , ak} and B = {b1, . . . , bl} be two finite subsets of Z with a1 < · ·...