This chapter discusses properties of univalent functions from the unit disc whose images are contained in the complex plane. The topics are chosen based on the needs of this book rather than their intrinsic relevance in the theory of univalent functions. The chapter begins with the No Koebe Arcs Theorem, which leads to results about pre-images of slits under univalent maps. It then presents the Koebe Distortion Theorems and concludes with the Carathéodory Kernel Convergence Theorem for families of univalent functions. Section 3.1 explores univalent functions and simply connected domains. A univalent function $ f : D \to C $ is holomorphic and injective, mapping the unit disc $ D $ to a simply connected domain $ \Omega $. The Riemann Mapping Theorem states that every simply connected domain $ \Omega \subset C $ (excluding $ C $ itself) has a unique univalent function $ f : D \to C $ mapping $ D $ onto $ \Omega $ with $ f(0) = w_0 $ and $ f'(0) > 0 $. This implies that simply connected domains and univalent functions from $ D $ are essentially the same. A key lemma allows restricting attention to bounded univalent functions. It shows that any univalent function $ f $ can be expressed in terms of a bounded univalent function $ h $, a point $ w_0 $, and a point $ z_0 $. A useful sufficient condition for a holomorphic function to be univalent is provided by the Noshiro-Warschawski Theorem. It states that if $ f $ is holomorphic on a convex domain $ \Delta $ and $ \text{Re} f'(z) > 0 $ for all $ z \in \Delta $, then $ f $ is univalent in $ \Delta $.This chapter discusses properties of univalent functions from the unit disc whose images are contained in the complex plane. The topics are chosen based on the needs of this book rather than their intrinsic relevance in the theory of univalent functions. The chapter begins with the No Koebe Arcs Theorem, which leads to results about pre-images of slits under univalent maps. It then presents the Koebe Distortion Theorems and concludes with the Carathéodory Kernel Convergence Theorem for families of univalent functions. Section 3.1 explores univalent functions and simply connected domains. A univalent function $ f : D \to C $ is holomorphic and injective, mapping the unit disc $ D $ to a simply connected domain $ \Omega $. The Riemann Mapping Theorem states that every simply connected domain $ \Omega \subset C $ (excluding $ C $ itself) has a unique univalent function $ f : D \to C $ mapping $ D $ onto $ \Omega $ with $ f(0) = w_0 $ and $ f'(0) > 0 $. This implies that simply connected domains and univalent functions from $ D $ are essentially the same. A key lemma allows restricting attention to bounded univalent functions. It shows that any univalent function $ f $ can be expressed in terms of a bounded univalent function $ h $, a point $ w_0 $, and a point $ z_0 $. A useful sufficient condition for a holomorphic function to be univalent is provided by the Noshiro-Warschawski Theorem. It states that if $ f $ is holomorphic on a convex domain $ \Delta $ and $ \text{Re} f'(z) > 0 $ for all $ z \in \Delta $, then $ f $ is univalent in $ \Delta $.