This　paper　surveys　in　a　tutorial　fashion　the　recent　history　of　universal　learning　machines　starting　with　the　multilayer　perceptron.　The　big　push　in　recent　years　has　been　on　the　design　of　universal　learning　machines　using　optimization　methods　linear　in　the　parameters,　such　as　the　Echo　State　Network,　the　Extreme　Learning　Machine　and　the　Kernel　Adaptive　filter.　We　call　this　class　of　learning　machines　convex　universal　learning　machines　or　CULMs.　The　purpose　of　the　paper　is　to　compare　the　methods　behind　these　CULMs,　highlighting　their　features　using　concepts　of　vector　spaces　(i.e.　basis　functions　and　projections),　which　are　easy　to　understand　by　the　computational　intelligence　community.　We　illustrate　how　two　of　the　CULMs　behave　in　a　simple　example,　and　we　conclude　that　indeed　it　is　practical　to　create　universal　mappers　with　convex　adaptation,　which　is　an　improvement　over　backpropagation.