Toeplitz matrices and operators constitute one of the most important and widely studied classes of non-self-adjoint operators. In this talk we consider truncated Toeplitz operators, a natural generalisation of finite Toeplitz matrices. They appear in various contexts, such as the study of finite interval convolution equations, signal processing, control theory, diffraction problems, hydrodynamics, elasticity, and they play a fundamental role in the study of complex symmetric operators. We will focus mainly on their invertibility and Fredholmness properties, showing in particular that they are equivalent after extension to block Toeplitz operators, and how this can be used to study the spectra of several classes of truncated Toeplitz operators.