In this article, we discuss the methodology based on Carleman estimates concerning the unique continuation and inverse problems of determining spatially varying coefficients. First as retrospective views we refer to main works by that methodology starting from the pioneering work by Bukhgeim and Klibanov published in 1981. Then as one possible object of the application of this methodology, we start to consider compressible fluid flows and we prove conditional stability estimates for an inverse source problem and the continuation of solutions from a part of lateral boundary. We apply two types of Carleman estimates respectively with a weight function which is quadratic in time and a weight function which rapidly decays at the end points of the time interval. © 2020, Springer Nature Singapore Pte Ltd.