In this thesis, we work with empirical likelihood functions and hybrid combinations of these with parametric likelihoods. We state and prove an analogue to Wilks theorem for the empirical likelihood function. In addition, we derive an alternative characterization of the map, and use it to reformulate maximization of the empirical likelihood function as an M-estimation problem. We also work with the hybrid likelihood function, a combination of empirical and parametric likelihoods. We prove a profiling result for this map and investigate the case of possible model misspecification. The limit distribution of the maximizer of the hybrid likelihood function is derived in this situation. In addition, we define a focused information criterion for hybrid likelihood and use it to propose a method for selecting the tuning parameters involved in the definition of the hybrid likelihood function.