Modern astronomical experiments offer many exciting challenges in the field of statistical signal processing. This paper gives an introduction to the problem of imaging the cosmic microwave background (CMB). The CMB is a ``fossil'' electromagnetic radiation ubiquitous in the universe since early times; its measurement and analysis currently are very active research topics in cosmology. The first part of this paper is a basic introduction to the physics behind the CMB and to its place in Big Bang theory: we describe the origin of the CMB and outline its relevance to cosmological models and the current experimental efforts. Accurately measuring and analyzing the CMB is a daunting task. In particular, CMB measurements are contaminated by other components from various origins. These components ---some of them also of cosmological interest--- can be separated by combining the sky maps made at several frequencies. This is the component separation problem which is a classic problem of statistical signal processing. The second part of this paper reviews specific aspects of the component separation problem in the context of CMB analysis.