Standard models for simple metals and insulators often fail for systems based on elements with unstable d- or f-electron shells, where strong electronic correlations can generate new and unexpected states of matter. Such a scenario can often be induced when a magnetic phase transition is tuned to absolute zero temperature by an external control parameter such as chemical composition, pressure or magnetic field. At the resulting quantum critical point (QCP), emergent phenomena, such as unconventional superconductivity and novel magnetic phases are frequently observed. The temperature and energy dependences of the physical properties are also found to deviate from expectations for a simple Fermi liquid. This “non-Fermi-liquid” (NFL) behavior is commonly manifested as weak power laws and logarithmic divergences in the physical properties at low temperatures and is often found in a V-shaped region near a QCP, which has become the “classic” QCP phase diagram. However, there is also a growing number of materials where the NFL behavior either occurs far away from the QCP, within an ordered phase, or may not be associated with any putative QCP. Thus, after nearly 20 years of research, it remains unknown whether NFL physics is universal, or if a multitude of unique subclasses exist. In this article, we review research that has primarily been carried out in our laboratory on systems that exhibit NFL behavior that does not conform to the “classic” QCP scenario.