Supply function equilibrium (SFE) models strategic offering in oligopolistic wholesale electricity auctions. In the supply function game, the strategies are curves giving the output levels offered at different prices. Generators submit their offer curves to the system operator, knowing the demand level only in distribution. A Nash equilibrium in these curves is found when each generator's offer is optimal, given the other offers, the demand distribution, and possible constraints and side payments. The theory for pool-market SFE with uniform pricing is well developed. This thesis explores several variations on SFE models. SFE models that include side payments arising from contracting arrangements or vertical integration between generators and retailers are developed. We find that forward sales through contracts for differences, options, and fixed price variable volume contracts all lead to lower wholesale prices, though each type of contract has a distinct effect on the shape of the price distribution. We consider non-uniform pricing rules, with an application to transmission pricing. The analysis is also extended to generators who act as price takers. The equilibrium conditions for SFE take the form of differential equations. Under appropriate assumptions on cost and demand functions, we obtain a planar autonomous system of ordinary differential equations for the SFE over a single lossless constrained transmission line. Computational methods are developed to handle monotonicity constraints on the supply functions, and then applied to topical electricity market policy questions on the competitive effects of: the rise of wind generation, futures and options trading, financial transmission rights, and asset swaps - real and virtual. Finally, a supply function equilibrium model is constructed for an auction in which multiple products are settled simultaneously, in which bids and offers are allowed to express cross-elasticities in demand and cost. With these new computational methods, it becomes possible to more accurately model the wholesale electricity market, and derive new insights into questions of market power and market design.