In this paper we discussed mechanism design and properties of social choice functions. First chapter studies theory of screening. Seller seeks to sell the good to one buyer and tries to find an optimal selling procedure to do so. We assume quasi linear utility functions i.e. these who are additively separable and risk neutral in money. We start by pricing a single indivisible good. We define buyer’s type as buyer’s valuation of the good that is not known to the seller. We wonder if picking a price \(p\) of the good is optimal selling strategy. We start by defining direct mechanism and state revelation principle which shows us that without loss of generality we can restrict our attention to these mechanisms. We introduce incentive compatibility and individual rationality constraints. We come to a conclusion that fixing a price really is best what seller can do. Further we turn to infinitely divisible good. Again we define direct mechanism in such environment and by using analogous propositions to the ones in previous section we get selling procedure that maximizes expected profit. In the end of this chapter we mention bundling where seller has two distinct indivisible goods, good \(A\) and good \(B\). We conclude that seller offers the buyer lower price if he buys goods as a bundle than if he had bought them separately. That is surprising considering that, from buyer’s point of view, goods are entirely unrelated. In the next chapter we give examples of mechanisms where there is one seller and two potential buyers. The highest bidder is declared the winner and he pays to the seller either the highest bid or the second highest. We implement these two cases by direct and indirect mechanisms. In implementation by indirect mechanisms we have famous first price and second price auctions with sealed bids. The last chapter studies non-transferable utility. We abandon assumptions about additive separability and risk neutrality. There is a finite set of agents \(I = \{1; 2; \dots, N\}\) which have to choose one alternative from a finite set \(A\) of mutually exclusive alternatives. Each agent has a preference relation \(R_i\) over \(A\). Again we define direct mechanism which, in this case, we call social choice function. Also, we define dominant strategy incentive compatibility and dictatorship. We state Gibbard Satterthwaite theorem, essential theorem of this paper. It states that if we assume that \(A\) has at least three elements and that direct mechanism \(f\) is surjective, then \(f\) is a dominant strategy incentive compatible if and only if it is dictatorial. Using a numerous auxiliary propositions we expose proof of this theorem in detail. For the end we cite possible relaxation of stringent requirements of GS theorem by restricting the domain. We introduce single peaked preference relations. In case of these preferences and same assumptions as in GS theorem, unlike previous result, there are direct mechanisms which are dominant strategy incentive compatible but not dictatorial.