It is difficult to make materials for applications that require highly specific microstructures. Consider the problem of making braids of micrometer-scale fibers for electronic and medical applications. In these applications, one must control the pattern of the braid (called the braid topology), which determines the material properties. Large and complex industrial machines have been developed to make various braid topologies at scale, but these machines are unsuitable for micrometer-scale fibers, which break too easily. I will describe an alternative fabrication method that relies on capillary forces, which arise when a floating object deforms a liquid interface. I will demonstrate how simple 3D-printed objects, which we call “capillary machines,” can grab floating objects and move them in programmed patterns. I will explain how we use the mathematics of braids to design the machines, and I will show how the machines can braid and weave micrometer-scale fibers -- as well as manipulate colloidal particles.