Histone lysine-specific demethylase 1 (LSD1/KDM1A) was first identified in 2004 as an epigenetic enzyme able to demethylate specific lysine residues of histone H3, namely H3K4me1/2 and H3K9me1/2, using FAD as the cofactor. It is ubiquitously overexpressed in many types of cancers (breast, gastric, prostate, hepatocellular, and esophageal cancer, acute myeloid leukemia, and others) leading to block of differentiation and increase of proliferation, migration and invasiveness at cellular level. LSD1 inhibitors can be grouped in covalent and non-covalent agents. Each group includes some hybrid compounds, able to inhibit LSD1 in addition to other target(s) at the same time (dual or multitargeting compounds). To date, 9 LSD1 inhibitors have entered clinical trials, for hematological and/or solid cancers. Seven of them (tranylcypromine, iadademstat (ORY-1001), bomedemstat (IMG-7289), GSK-2879552, INCB059872, JBI-802, and Phenelzine) covalently bind the FAD cofactor, and two are non-covalent LSD1 inhibitors [pulrodemstat (CC-90011) and seclidemstat (SP-2577)]. Another TCP-based LSD1/MAO-B dual inhibitor, vafidemstat (ORY-2001), is in clinical trial for Alzheimer’s diseases and personality disorders. The present review summarizes the structure and functions of LSD1, its pathological implications in cancer and non-cancer diseases, and the identification of LSD1 covalent and non-covalent inhibitors with different chemical scaffolds, including those involved in clinical trials, highlighting their potential as potent and selective anticancer agents.