The design of membrane-based constructs with multiple compartments is of increasing importance given their potential applications as microreactors, as artificial cells in synthetic-biology, as simplified cell models, and as drug delivery vehicles. The emergence of droplet microfluidics as a tool for their construction has allowed rapid scale-up in generation throughput, scale-down of size, and control over gross membrane architecture. This is true on several levels: size, level of compartmentalisation and connectivity of compartments can all be programmed to various degrees. This tutorial review explains and explores the reasons behind this. We discuss microfluidic strategies for the generation of a family of compartmentalised systems that have lipid membranes as the basic structural motifs, where droplets are either the fundamental building blocks, or are precursors to the membrane-bound compartments. We examine the key properties associated with these systems (including stability, yield, encapsulation efficiency), discuss relevant device fabrication technologies, and outline the technical challenges. In doing so, we critically review the state-of-play in this rapidly advancing field.