This　paper　is　devoted　to　the　study　and　analysis　of　the　maximum　signal-to-noise　ratio　(SNR)　filters　for　noise　reduction　both　in　the　time　and　short-time　Fourier　transform　(STFT)　domains　with　one　single　microphone　and　multiple　microphones.　In　the　time　domain,　we　show　that　the　maximum　SNR　filters　can　significantly　increase　the　SNR　but　at　the　expense　of　tremendous　speech　distortion.　As　a　consequence,　the　speech　quality　improvement,　measured　by　the　perceptual　evaluation　of　speech　quality　(PESQ)　algorithm,　is　marginal　if　any,　regardless　of　the　number　of　microphones　used.　In　the　STFT　domain,　the　maximum　SNR　filters　are　formulated　by　considering　the　interframe　information　in　every　frequency　band.　It　is　found　that　these　filters　not　only　improve　the　SNR,　but　also　improve　the　speech　quality　significantly.　As　the　number　of　input　channels　increases　so　is　the　gain　in　SNR　as　well　as　the　speech　quality.　This　demonstrates　that　the　maximum　SNR　filters,　particularly　the　multichannel　ones,　in　the　STFT　domain　may　be　of　great　practical　value.