Wavelet theory provides a unified framework for various signal processing techniques, including multiresolution analysis, subband coding, and wavelet series expansions. It covers both continuous and discrete-time cases and offers general techniques applicable to many signal processing tasks. The Wavelet Transform (WT) is particularly useful for analyzing non-stationary signals, offering an alternative to the Short-Time Fourier Transform (STFT) by using variable window sizes. The WT introduces the concept of scale as an alternative to frequency, leading to a time-scale representation. Different types of wavelet transforms, such as the Continuous Wavelet Transform (CWT), Wavelet Series expansion, and Discrete Wavelet Transform (DWT), are used depending on the application. The DWT uses multirate signal processing techniques and is related to subband coding in speech and image compression. Wavelet theory has been developed as a unifying framework, with roots in early mathematical and physical concepts. The "French school" of researchers, including Morlet, Grossmann, and Meyer, established strong mathematical foundations and named their work "Wavelets." The theory has grown rapidly, with significant contributions from various fields. The paper serves as a review and tutorial, covering the main definitions and properties of wavelet transforms, their connections to different fields, and their applications in signal processing. The Short-Time Fourier Transform (STFT) provides a time-frequency representation of signals but has fixed resolution. The Continuous Wavelet Transform (CWT) allows for variable resolution, making it suitable for analyzing non-stationary signals. The CWT uses scaled versions of a prototype wavelet, leading to a time-scale representation. The Discrete Wavelet Transform (DWT) is used for discrete-time signals and is related to subband coding. Wavelet analysis involves decomposing signals into wavelet basis functions, which are scaled and shifted versions of a prototype wavelet. Wavelet analysis results in wavelet coefficients that indicate how close a signal is to a particular basis function. This allows for the representation of any general signal as a decomposition into wavelets. The reconstruction of a signal from wavelet coefficients is possible under certain conditions, such as the use of tight frames or orthonormal bases. The wavelet transform can be related to the Wigner-Ville distribution, and there are strong links between wavelet transforms and time-frequency energy distributions. Wavelet frames provide a general framework that balances redundancy and restrictions on basis functions for reconstruction. The Discrete Wavelet Transform (DWT) is used for discrete-time signals and is related to subband coding. The DWT uses multirate signal processing techniques and is related to subband coding in speech and image compression. The paper also discusses the use of complex wavelet transforms and their visualization through phasemagrams. The Discrete Wavelet Transform is a key tool in signal processing, offering efficient and effective methods for analyzing and reconstructing signals.Wavelet theory provides a unified framework for various signal processing techniques, including multiresolution analysis, subband coding, and wavelet series expansions. It covers both continuous and discrete-time cases and offers general techniques applicable to many signal processing tasks. The Wavelet Transform (WT) is particularly useful for analyzing non-stationary signals, offering an alternative to the Short-Time Fourier Transform (STFT) by using variable window sizes. The WT introduces the concept of scale as an alternative to frequency, leading to a time-scale representation. Different types of wavelet transforms, such as the Continuous Wavelet Transform (CWT), Wavelet Series expansion, and Discrete Wavelet Transform (DWT), are used depending on the application. The DWT uses multirate signal processing techniques and is related to subband coding in speech and image compression. Wavelet theory has been developed as a unifying framework, with roots in early mathematical and physical concepts. The "French school" of researchers, including Morlet, Grossmann, and Meyer, established strong mathematical foundations and named their work "Wavelets." The theory has grown rapidly, with significant contributions from various fields. The paper serves as a review and tutorial, covering the main definitions and properties of wavelet transforms, their connections to different fields, and their applications in signal processing. The Short-Time Fourier Transform (STFT) provides a time-frequency representation of signals but has fixed resolution. The Continuous Wavelet Transform (CWT) allows for variable resolution, making it suitable for analyzing non-stationary signals. The CWT uses scaled versions of a prototype wavelet, leading to a time-scale representation. The Discrete Wavelet Transform (DWT) is used for discrete-time signals and is related to subband coding. Wavelet analysis involves decomposing signals into wavelet basis functions, which are scaled and shifted versions of a prototype wavelet. Wavelet analysis results in wavelet coefficients that indicate how close a signal is to a particular basis function. This allows for the representation of any general signal as a decomposition into wavelets. The reconstruction of a signal from wavelet coefficients is possible under certain conditions, such as the use of tight frames or orthonormal bases. The wavelet transform can be related to the Wigner-Ville distribution, and there are strong links between wavelet transforms and time-frequency energy distributions. Wavelet frames provide a general framework that balances redundancy and restrictions on basis functions for reconstruction. The Discrete Wavelet Transform (DWT) is used for discrete-time signals and is related to subband coding. The DWT uses multirate signal processing techniques and is related to subband coding in speech and image compression. The paper also discusses the use of complex wavelet transforms and their visualization through phasemagrams. The Discrete Wavelet Transform is a key tool in signal processing, offering efficient and effective methods for analyzing and reconstructing signals.