The structure studied in the paper is the fuzzy metric space (a modified definition of the statistical metric space), which is a triple \((X, M, *)\), where \(X\) is the universe, \(*\) is a triangular norm and \(M:X^2 \times (0, \infty) \to [0,1]\) with properties usual for fuzzy metric spaces. Special attention is given to strong fuzzy metric spaces, i.e. spaces, where \(M\) is continuous in its first variable. Problems of best approximation are studied in these spaces. The author describes the closure of a set in terms of \(M\) and proves the theorem on the existence of best approximation.