Leishmaniasis is a tropical infectious disease caused by the protozoan Leishmania parasite. The disease is transmitted by female sand flies and, depending on the infecting parasite species, causes either cutaneous (stigmatizing skin lesions), mucocutaneous (destruction of mucous membranes of nose, mouth and throat) or visceral disease (a potentially fatal infection of liver, spleen and bone marrow). Although more than 1 million new cases occur annually, chemotherapeutic options are limited and their efficacy is jeopardized by increasing treatment failure rates and growing drug resistance. To delay the emergence of resistance to existing and new drugs, elucidating the currently unknown causes of variable drug efficacy (related to parasite susceptibility, host immunity and drug pharmacokinetics) and improved use of genotypic and phenotypic tools to define, measure and monitor resistance in the field are critical. This review highlights recent progress in our understanding of drug action and resistance in Leishmania, ongoing challenges (including setbacks related to the COVID-19 pandemic) and provides an overview of possible strategies to tackle this public health challenge.