This paper considers the solution of the bioelectromagnetic inverse problem with particular emphasis on focal compact sources that are likely to arise in epileptic data. Two linear inverse methods are proposed and evaluated in simulations. The first method belongs to the class of distributed inverse solutions, capable of dealing with multiple simultaneously active sources. This solution is based on a Local Auto Regressive Average (LAURA) model. Since no assumption is made about the number of activated sources, this approach can be applied to data with multiple sources. The second method, EPIFOCUS, assumes that there is only a single focal source. However, in contrast to the single dipole model, it allows the source to have a spatial extent beyond a single point and avoids the non-linear optimization process required by dipole fitting. The performance of both methods is evaluated with synthetic data in noisy and noise free conditions. The simulation results demonstrate that LAURA and EPIFOCUS increase the number of sources retrieved with zero dipole localization error and produce lower maximum error and lower average error compared to Minimum Norm, Weighted Minimum Norm and Minimum Laplacian (LORETA). The results show that EPIFOCUS is a robust and powerful tool to localize focal sources. Alternatives to localize data generated by multiple sources are discussed. A companion paper (Lantz et al. 2001, this issue) illustrates the application of LAURA and EPIFOCUS to the analysis of interictal data in epileptic patients.