Substance addiction (or drug addiction) is defined as a neuropsychiatric disorder characterized by a recurring desire to continue taking the drug despite harmful consequences. It is hypothesized to follow a heuristic framework that involves three stages (binge/intoxication, withdrawal/negative effect, and preoccupation/anticipation). Non-substance addiction (or behavioral addiction) mainly includes pathological gambling, food addiction, internet addiction, and mobile phone addiction. Their definition is similar to drug addiction, but differs from each other in specific domains. In this review, we have three main objectives. First, we aim to provide a brief overview of past and current definitions of substance and non-substance addiction. Second, we also touch on the topic of diagnosing drug addiction and non-drug addiction. Finally, this review focuses on furthering the understanding of the key concepts needed for a foundation to study the biological and psychological underpinnings of addiction disorders.