In this chapter, we focus on Carleman estimates for operators with smooth coefficients for the principal part and bounded coefficients for lower-order terms. We examine elliptic operators with complex-valued coefficients and operators with real-valued principal parts. We take an accessible approach to Carleman estimates, laying the groundwork for the later use of pseudodifferential methods. The focus is on differential operator symbols and their role in Carleman estimates with the large parameter introduced in the previous chapter. We introduce essential concepts such as the Poisson bracket, Sobolev spaces with large parameters, and the commutator of the self-adjoint and skew-adjoint parts of an operator. These lead to the proof of the localized Gårding inequality, a key tool for Carleman estimates. In subsequent sections, we develop a Carleman estimate with weight exponent and define a conjugate operator. We also provide approximation formulas for differential operators depending on a large parameter and explore commutators. These results are applied to establish Carleman estimates for elliptic operators and operators with real-valued principal parts. The chapter also covers the application of Carleman estimates to uniqueness in the Cauchy problem, introduces pseudo-convex surfaces, and briefly discusses pseudodifferential operators and the Sharp Gårding Inequality, illustrating their usefulness in proving Carleman estimates.