A magic \(( m, n)\) rectangle is an \(m\times n\) matrix with entries \(1,\dots, mn\) for which all the row sums are the same and all the column sums are the same. A modular magic \((m, n)\) rectangle is an \(m\times n\) matrix with entries \(1,\dots, mn\) for which all the row sums are congruent modulo \(mn\) and all the column sums are congruent modulo \(mn\). It is shown that the existence criteria for modular magic rectangles are virtually the same as for magic rectangles. Two classes of modular magic rectangles are studied: pseudomagic and complete magic rectangles. Three open problems are mentioned.