We present a detailed analysis of four of the most widely used radio source-finding packages in radio astronomy, and a program being developed for the Australian Square Kilometer Array Pathfinder telescope. The four packages: sextractor, sfind, imsad and selavy are shown to produce source catalogues with high completeness and reliability. In this paper we analyse the small fraction (∼1 per cent) of cases in which these packages do not perform well. This small fraction of sources will be of concern for the next generation of radio surveys which will produce many thousands of sources on a daily basis, in particular for blind radio transients surveys. From our analysis we identify the ways in which the underlying source-finding algorithms fail. We demonstrate a new source-finding algorithm aegean, based on the application of a Laplacian kernel, which can avoid these problems and can produce complete and reliable source catalogues for the next generation of radio surveys.