In this talk, I'll first give an overview of communication complexity of combinatorial auctions, and then discuss some recent results. Combinatorial auctions center around the following problem: There is a set M of m items, and N of n bidders. Each bidder i has a valuation function v_i from subsets of M to real numbers. The goal is to partition the items into S_1,\ldots, S_n maximizing \sum_i v_i(S_i).

A central problem of study is understanding the communication required between the bidders (who each know their own valuation, but not the others).

- Algorithm Design version: all parties honestly participate in whatever protocol is proposed.

- Mechanism Design version: The designer is allowed to charge price p_i to agent i, if desired. Party i will behave in whatever way they believe maximizes v_i(S_i) - p_i.

A sample of results I'll aim to discuss (at varying levels of detail):

- Tight bounds on achievable approximation guarantees for algorithms.

- Maximal-in-range auctions: tight bounds on the approximation guarantees achievable by "VCG-like" ideas.

- Separations between approximation guarantees achievable by algorithms vs. mechanisms.