This little book is a revised and expanded version of one I wrote for the "VIII Latin American School of Mathematics" [29], in Portuguese, based on which I have periodically taught a topics course over the last 18 years. As the - graduate students tle suggests, it is an introductory text. It is addressed to of mathematics in the area of differential equations/nonlinear analysis and to mathematicians in other areas who would like to have a first exposure to so called variational methods and their applications to PDEs and ODEs. Afterwards, the reader can choose from some excellent and more compreh- sive texts, which already exist in the literature but require somewhat more maturity in the area. We present a cross section of the area of variational methods, with a m- imum (no "pun" intended) of material, but clearly illustrating through one or two examples each of the results that we have chosen to present. So, besides the first motivating chapter and an appendix, there are only ten short chapters (with three or, at most, four sections each) through which the reader is quickly exposed to a few basic aspects of the beautiful area of variational methods and applications to differential equations. In fact, the reader may initially skip some of the more technical proofs of the main theorems, concentrating instead on the applications that are given.
