This chapter focuses on univalent functions, which are holomorphic and injective functions from the unit disc \(\mathbb{D}\) to the complex plane \(\mathbb{C}\). The chapter begins by proving the No Koebe Arcs Theorem, which leads to several results about pre-images of slits via univalent maps. It then presents the Koebe Distortion Theorems and discusses families of univalent functions, culminating in the proof of the Carathéodory Kernel Convergence Theorem. The chapter also delves into the relationship between univalent functions and simply connected domains. It states that if \(\Omega\) is a simply connected domain in \(\mathbb{C}\) (excluding \(\mathbb{C}\) itself), then for every \(w_0 \in \Omega\), there exists a unique univalent function \(f : \mathbb{D} \to \mathbb{C}\) such that \(f(\mathbb{D}) = \Omega\) and \(f'(0) = w_0\). This function is called a Riemann map of \(\Omega\). A key lemma is introduced, which allows for the restriction of attention to bounded univalent functions. The lemma states that any univalent function \(f : \mathbb{D} \to \mathbb{C}\) can be expressed in a specific form involving a never-vanishing bounded univalent function \(h\) and constants \(w_0\) and \(z_0\). Finally, the chapter presents Noshiro-Warschawski's Theorem, which provides a sufficient condition for a holomorphic function to be univalent in a convex domain \(\Delta\). The theorem asserts that if \(\operatorname{Re} f'(z) > 0\) for all \(z \in \Delta\), then \(f\) is univalent in \(\Delta\). The proof of this theorem involves showing that the integral of the real part of \(f'\) over any segment connecting two points in \(\Delta\) cannot be zero, thus proving injectivity.This chapter focuses on univalent functions, which are holomorphic and injective functions from the unit disc \(\mathbb{D}\) to the complex plane \(\mathbb{C}\). The chapter begins by proving the No Koebe Arcs Theorem, which leads to several results about pre-images of slits via univalent maps. It then presents the Koebe Distortion Theorems and discusses families of univalent functions, culminating in the proof of the Carathéodory Kernel Convergence Theorem. The chapter also delves into the relationship between univalent functions and simply connected domains. It states that if \(\Omega\) is a simply connected domain in \(\mathbb{C}\) (excluding \(\mathbb{C}\) itself), then for every \(w_0 \in \Omega\), there exists a unique univalent function \(f : \mathbb{D} \to \mathbb{C}\) such that \(f(\mathbb{D}) = \Omega\) and \(f'(0) = w_0\). This function is called a Riemann map of \(\Omega\). A key lemma is introduced, which allows for the restriction of attention to bounded univalent functions. The lemma states that any univalent function \(f : \mathbb{D} \to \mathbb{C}\) can be expressed in a specific form involving a never-vanishing bounded univalent function \(h\) and constants \(w_0\) and \(z_0\). Finally, the chapter presents Noshiro-Warschawski's Theorem, which provides a sufficient condition for a holomorphic function to be univalent in a convex domain \(\Delta\). The theorem asserts that if \(\operatorname{Re} f'(z) > 0\) for all \(z \in \Delta\), then \(f\) is univalent in \(\Delta\). The proof of this theorem involves showing that the integral of the real part of \(f'\) over any segment connecting two points in \(\Delta\) cannot be zero, thus proving injectivity.