Constructed wetland (CW) is a popular sustainable best management practice for treating different wastewaters. While there are many articles on the removal of pollutants from different wastewaters, a comprehensive and critical review on the removal of pollutants other than nutrients that occur in agricultural field runoff and wastewater from animal facilities, including pesticides, insecticides, veterinary medicine, and antimicrobial-resistant genes are currently unavailable. Consequently, this paper summarized recent findings on the occurrence of such pollutants in the agricultural runoff water, their removal by different wetlands (surface flow, subsurface horizontal flow, subsurface vertical flow, and hybrid), and removal mechanisms, and analyzed the factors that affect the removal. The information is then used to highlight the current research gaps and needs for resilient and sustainable treatment systems. Factors, including contaminant property, aeration, type, and design of CWs, hydraulic parameters, substrate medium, and vegetation, impact the removal performance of the CWs. Hydraulic loading of 10–30 cm/d and hydraulic retention of 6–8 days were found to be optimal for the removal of agricultural pollutants from wetlands. The pollutants in agricultural wastewater, excluding nutrients and sediment, and their treatment utilizing different nature-based solutions, such as wetlands, are understudied, implying the need for more of such studies. This study reinforced the notion that wetlands are effective for treating agricultural wastewater (removal >90%) but several research questions remain unanswered. More long-term research in the actual field utilizing environmentally relevant concentrations to seek actual impacts of weather, plants, substrates, hydrology, and other design parameters, such as aeration and layout of wetland cells on the removal of pollutants, are needed.