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For a long time, detection and parameter estimation methods for signal processing have relied on asymptotic statistics as the number n of observations of a population grows large comparatively to the population size N, i.e., n/N → ∞. Modern technological and societal advances now demand the study of sometimes extremely large populations and simultaneously require fast signal processing due to accelerated system dynamics. This results in not-so-large practical ratios n/N, sometimes even smaller than one. A disruptive change in classical signal processing methods has therefore been initiated in the past ten years, mostly spurred by the field of large-dimensional random matrix theory. The early works in random matrix theory for signal processing applications are, however, scarce and highly technical. This tutorial provides an accessible methodological introduction to the modern tools of random matrix theory and to the signal processing methods derived from them, with an emphasis on simple illustrative examples.