Due to its carcinogenic properties, the presence of formaldehyde in resins and other industrial products has been a subject of great concern in recent years. The presented review focuses on modern alternatives for the production of wood-based panels; i.e., substitutes for formaldehyde in the production of amino and phenolic resins, as well as novel hardeners for formaldehyde-free wood adhesives. Solutions in which formaldehyde in completely replaced are presented in this review. Recent advances indicate that it is possible to develop new formaldehyde-free systems of resins with compatible hardeners. The formaldehyde substitutes that have primarily been tested are glyoxal, glutaraldehyde, furfural, 5-hydroxymethylfurfural, and dimethoxyethanal. The use of such substitutes eliminates the problem of free formaldehyde emission originating from the resin used in the production of wood-based panels. However, these alternatives are mostly characterized by worse reactivity, and, as a result, the use of formaldehyde-free resins may affect the mechanical and strength properties of wood-based panels. Nonetheless, there are still many substantial challenges for the complete replacement of formaldehyde and further research is needed, especially in the field of transferring the technology to industrial practice.