Degradable and cell-compatible hydrogels can be designed to mimic the physical and biochemical characteristics of native extracellular matrices and provide tunability of degradation rates and related properties under physiological conditions. Hence, such hydrogels are finding widespread application in many bioengineering fields, including controlled bioactive molecule delivery, cell encapsulation for controlled three-dimensional culture, and tissue engineering. Cellular processes, such as adhesion, proliferation, spreading, migration, and differentiation, can be controlled within degradable, cell-compatible hydrogels with temporal tuning of biochemical or biophysical cues, such as growth factor presentation or hydrogel stiffness. However, thoughtful selection of hydrogel base materials, formation chemistries, and degradable moieties is necessary to achieve the appropriate level of property control and desired cellular response. In this review, hydrogel design considerations and materials for hydrogel preparation, ranging from natural polymers to synthetic polymers, are overviewed. Recent advances in chemical and physical methods to crosslink hydrogels are highlighted, as well as recent developments in controlling hydrogel degradation rates and modes of degradation. Special attention is given to spatial or temporal presentation of various biochemical and biophysical cues to modulate cell response in static (i.e., non-degradable) or dynamic (i.e., degradable) microenvironments. This review provides insight into the design of new cell-compatible, degradable hydrogels to understand and modulate cellular processes for various biomedical applications.