The field of research on cellular senescence experienced a rapid expansion from being primarily focused on in vitro aspects of aging to the vast territories of animal and clinical research. Cellular senescence is defined by a set of markers, many of which are present and accumulate in a gradual manner prior to senescence induction or are found outside of the context of cellular senescence. These markers are now used to measure the impact of cellular senescence on aging and disease as well as outcomes of anti-senescence interventions, many of which are at the stage of clinical trials. It is thus of primary importance to discuss their specificity as well as their role in the establishment of senescence. Here, the presence and role of senescence markers are described in cells prior to cell cycle arrest, especially in the context of replicative aging and in vivo conditions. Specifically, this review article seeks to describe the process of "cellular aging": the progression of internal changes occurring in primary cells leading to the induction of cellular senescence and culminating in cell death. Phenotypic changes associated with aging prior to senescence induction will be characterized, as well as their effect on the induction of cell senescence and the final fate of cells reviewed. Using published datasets on assessments of senescence markers in vivo, it will be described how disparities between quantifications can be explained by the concept of cellular aging. Finally, throughout the article the applicational value of broadening cellular senescence paradigm will be discussed.