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A problem of much interest is the estimation of the delay or shift between two signals or images. In this paper, we investigate the use of recently proposed [Yegnanarayana et.al., IEEE Trans. ASSP, Vol 32, 610, 1984] Group Delay Functions (GDFs) for estimating the delay between two signals. The two types of GDFs (one based on the Fourier phase and the other based on the Fourier magnitude) contain different types of information relevant to the delay estimation problem. The phase-based GDF contains the delay information whereas the magnitude based GDF contains the Signal-to-Noise Ratio (SNR) information. We propose an adaptive delay estimation technique that utilizes both types of information. This adaptive technique weights the delay estimates from the high-SNR regions more than the delay estimates from the low-SNR regions. Since the GDFs take on a continuum of values, the proposed technique does not distinguish between integer delays and fractional delays. Thus the GDF-based delay estimation procedure is inherently capable of subpixel delay estimation. Simulation results are also included to illustrate the capabilities of this procedure.