Numerous surgical procedures are daily performed worldwide to replace and repair damaged tissue. Tissue engineering is the field devoted to the regeneration of damaged tissue through the incorporation of cells in biocompatible and biodegradable porous constructs, known as scaffolds. The scaffolds act as host biomaterials of the incubating cells, guiding their attachment, growth, differentiation, proliferation, phenotype, and migration for the development of new tissue. Furthermore, cellular behavior and fate are bound to the biodegradation of the scaffold during tissue generation. This article provides a critical appraisal of how key biomaterial scaffold parameters, such as structure architecture, biochemistry, mechanical behavior, and biodegradability, impart the needed morphological, structural, and biochemical cues for eliciting cell behavior in various tissue engineering applications. Particular emphasis is given on specific scaffold attributes pertaining to skin and brain tissue generation, where further progress is needed (skin) or the research is at a relatively primitive stage (brain), and the enumeration of some of the most important challenges regarding scaffold constructs for tissue engineering.