Many challenging issues arise in the newly deregulated competitive electric power markets worldwide. Instead of centralized decision-making in a monopoly environment as in the past, many parties with different goals are now involved and competing in the market. The information available to a party may be limited, regulated, and received with time delay, and decisions made by one party may influence the decision space and well-being of others. These difficulties are compounded by the underlying uncertainties inherent in the system such as the demand for electricity, fuel prices, outages of generators and transmission lines, tactics by certain market participants, etc. Consequently the market is full of uncertainty and risk. Key questions to be addressed include how to predict load and market clearing prices, how to consider other parties‘ decisions in deciding one‘s own bids, and how to manage uncertainty and risk. Since finding an optimal solution to a traditional unit commitment problem is NP-hard even without considering multiple parties and uncertainties, it may be more practical to know which decision is good with confidence rather than looking for an optimal solution. For an energy supplier, bidding decisions are coupled with generation resource scheduling or unit commitment since generator characteristics and how they will be used to meet the accepted bids in the future have to be considered before bids are submitted. For example, if starting up a thermal unit is expected, the associated startup cost should somehow be configured in the bid curves. The decisions, however, can be quite subtle since generally startup costs are not part of a bid. Bidding decisions should therefore be carefully made by considering the anticipated MCP, system demand, generation and startup costs, and competitor‘ decisions. What further complicates the issue is that some of the information is not available, or will be available but with significant delays. In paper, two promising bidding strategies for power suppliers are discussed. The ordinal optimization method seeks ’’good enough‘‘ bids with high probability, and is an effective in handling market uncertainties with much reduced computational efforts. The basic idea of this method is to use a model to describe the influence of bidding strategies on the MCP. A nominal bid curve is obtained by solving optimal power generation for a given set of the MCPs within the Lagrangian relaxation framework. Then N bids are generated by perturbing the nominal bid curve. The ordinal optimization method is applied to form a good enough bid set S, which contains some good bids with high probability, by performing rough evaluation. The best bid is then searched and selected over S by solving generation scheduling or unit commitment problems within reasonable computational time. The game theoretic method aims at bidding and self-scheduling of a utility company in New England. The problem is investigated from the viewpoint of a particular utility bidder. The uncertainties caused by bids from other bidders and the ISO (Independent System Operator) bid selection process are explicitly considered. The problem is then solved within a reduced game theoretical framework, where the ISO has a closed-form solution for a given probabilistic description of the bids, and the utility‘s problem is solved by using Lagrangian relaxation. Although the two specific methods represent significant progress made thus far, the area is wide open for creative research to make the deregulated market a true success.