We provide an introduction to the use of spectral-element and adjoint methods in seismology. Following a brief review of the basic equations that govern seismic wave propagation, we discuss how these equations may be solved numerically based upon the spectral-element method (SEM) to address the forward problem in seismology. Examples of synthetic seismograms calculated based upon the SEM are compared to data recorded by global and regional seismographic networks. We also discuss the challenge of using the remaining differences between the data and the synthetic seismograms to constrain better Earth models and source descriptions. This leads naturally to adjoint methods, which provide a practical approach to this formidable computational challenge and enables seismologists to tackle the inverse problem.