Resumo: We present recent results for the KP-II equations with generalized dispersion terms, in two and three spacial dimensions, periodic only in the x variable. We will start by showing how the solutions to the linearized equations satisfy bilinear Strichartz-type estimates, which are independent of the dispersion. We then use these estimates to establish local well-posedness for the Cauchy problem associated to the equations for low regularity data, in the framework of Bourgain spaces. For certain ranges of dispersion, these local results are optimal. This is a joint work with Axel Grünrock and Mahendra Panthee.