This chapter explores wavelet design using regularity conditions and moments, which influence smoothness and frequency localization. It begins with the Haar scaling function which is the simplest wavelet basis and then introduces the Haar wavelet function, demonstrating its compact support and piecewise constant nature. The relationship between scaling and wavelet functions is then analyzed emphasizing how scaling functions generate wavelets through refinement relations. The concept of refinement relations is discussed, highlighting their role in constructing wavelets with desired properties. The chapter further examines Daubechies wavelets known for their compact support and maximum regularity for a given number of vanishing moments. Next, Coiflet wavelets are introduced, designed for improved symmetry and higher order moments. The chapter concludes with an overview of other common wavelets, providing insight into their mathematical construction and applications in signal processing.