Additive manufacturing (AM) is the process through which components/structures are produced layer-by-layer. In this context, 4D printing combines 3D printing with time so that this combination results in additively manufactured components that respond to external stimuli and, consequently, change their shape/volume or modify their mechanical properties. Therefore, 4D printing uses shape-memory materials that react to external stimuli such as pH, humidity, and temperature. Among the possible materials with shape memory effect (SME), the most suitable for additive manufacturing are shape memory polymers (SMPs). However, due to their weaknesses, shape memory polymer compounds (SMPCs) prove to be an effective alternative. On the other hand, out of all the additive manufacturing techniques, the most widely used is fused filament fabrication (FFF). In this context, the present paper aims to critically review all studies related to the mechanical properties of 4D-FFF materials. The paper provides an update state of the art showing the potential of 4D-FFF printing for different engineering applications, maintaining the focus on the structural integrity of the final structure/component.