Background: Malnutrition is a major global public health issue, especially for under five children and their mothers. Objective: This study examined the relationships between socio-economic and demographic characteristics and under- and over-nutrition in mothers and children of Bangladesh. Method: Using the Nationally representative Bangladesh Demographic and Health Survey (2017-18) data, we performed the multiple logistic regression analysis to assess the relationships between key outcome variables and predictors. The analysis included 8321 children aged 0-59 months and 7800 mothers aged 15-49 years. Findings: The prevalence of stunting, wasting, and overweight among children under five was 30.72%, 8.44%, and 2.21% respectively. About 2.74% of the children were both stunted and wasted, while 0.56% of the children were both stunted and overweight. The prevalence of underweight and overweight among mother was 14.09% and 26.35% respectively. Children who are stunted are significantly associated with the mother's BMI status, mothers' education, fathers' education, and wealth index, while children who are wasted are significantly associated with the mother's BMI status and the child's sex. A child's sex and birth order are significantly associated with children being overweight. Mother's age, mother's education, father's education, place of residence and wealth index are significant determinants for mothers being underweight, whereas mother's age, mother's occupation, father's education, father's occupation, place of residence and wealth index are significant determinants for mothers being overweight. Conclusion: In addition to essential nutrition interventions, it is vital to address the basic social-economic and demographic determinants.