The objective of the work is to study the well-posedness of the Kadomtsev-Petviashvili (KP) equation of type I. The KP equations of types I and II are two-dimensional generalizations of the Korteweg-de Vries equation, with opposite signs of the two-dimensional dispersion. Unlike the KP equation of the type II, the proof of the well-posedness of the initial value problem for the KP-I equation is essentially harder. In this work, the existence of a solution to the initial value problem for the KP-I equation with small initial data from an appropriate Sobolev space is proven by means of estimates of oscillatory integrals that give an approximation to the solution. The Sobolev space to which the initial conditions must belong involves derivatives of the total order no larger than 2.