It has been known since the 1930's that protons and neutrons, collectively called as nucleons, are not "point-like" elementary particles, but rather have a substructure. Today, we know from Quantum Chromodynamics (QCD) that nucleons are made from quarks and gluons, with gluons being the elementary force carriers for strong interactions. Quarks and gluons are collectively called as partons. The substructure of the nucleons can be described in terms of parton correlation functions such as Form Factors, (1D) Parton Distribution Functions (PDFs) and their 3D generalizations in terms of Transverse Momentum-dependent parton Distributions (TMDs) and Generalized Parton Distributions (GPDs). All these functions can be derived from the even more general Generalized Transverse Momentum-dependent Distributions (GTMDs). This dissertation promises to provide an insight into all these functions from the point of view of their accessibility in experiments, from model calculations, and from their direct calculation within lattice formulations of QCD. In the first part of this dissertation, we identify physical processes to access GTMDs. By considering the exclusive double Drell-Yan process, we demonstrate, for the very first time, that quark GTMDs can be measured. We also show that exclusive double-quarkonium production in nucleon-nucleon collisions is a direct probe of gluon GTMDs. In the second part of this dissertation, we shift our focus to the "parton quasi-distributions". Over the last few decades, lattice QCD extraction of the full x-dependence of the parton distributions has always been prohibited by the explicit time-dependence of the correlation functions. In 2013, there was a path-breaking proposal by X. Ji to calculate instead parton quasi-distributions (quasi-PDFs). The procedure of "matching" is a crucial ingredient in the lattice QCD extraction of parton distributions from the quasi-PDF approach. We address the matching for the twist-3 PDFs gT (x), e(x), and hL(x) for the very first time. We pay special attention to the challenges involved in the calculations due to the presence of singular zero-mode contributions. We also present the first-ever lattice QCD results for gT (x) and hL(x) and we discuss the impact of these results on the phenomenology. Next, we explore the general features of quasi-GPDs and quasi-PDFs in diquark spectator models. Furthermore, we address the Burkhardt-Cottingham-type sum rules for the relevant light-cone PDFs and quasi-PDFs in a model-independent manner and also check them explicitly in perturbative model calculations. The last part of this dissertation focuses on the extraction g1T (x,~k2⊥) TMD for the very first time from experimental data using Monte Carlo techniques. This dissertation therefore unravels different aspects of the distribution functions from varied perspectives.