In this paper, we show that of all graphs of order n with matching number β, the graphs with maximal spectral radius are K n if n = 2 β or 2 β + 1; K 2 β + 1 ∪ K n - 2 β - 1 ¯ if 2 β + 2 ⩽ n < 3 β + 2; K β ⋁ K n - β ¯ or K 2 β + 1 ∪ K n - 2 β - 1 ¯ if n = 3 β + 2; K β ⋁ K n - β ¯ if n > 3 β + 2, where K t ¯ is the empty graph on t vertices.