Periodic structures have received conspicuous attention in recent years due to their inherent dynamic (phononic) characteristics that impart them the capability to function as mechanical filters or waveguides. In this regard, nonlinear periodic structures feature tunable phononic characteristics, and the ability to support unique features seldom observed in their linear counterparts. Therefore, the characteristics of propagating wave packets in nonlinear periodic structures are studied with the motive of leveraging the unique features associated with nonlinearity to achieve adaptive and tunable spatial wave manipulation. Specifically, the interplay of dispersion and nonlinearity is characterized for weakly nonlinear systems capable of supporting a wide variety of deformation patterns and crystal responses. To this end, multiple scale analysis of a simple nonlinear periodic structure is carried out under the assumption of small, but non-negligible nonlinear terms to establish the foundation for numerical exploration of a number of complex crystal configurations. It is observed that the primary effect of the interplay between nonlinearity and dispersion manifests as a modulation of the envelope of propagating wave packets. In the case of nonlinearity being represented by an equivalent quadratic nonlinear term, the envelope modulation is observed to result in the generation of a non-harmonic component in the response. This non-harmonic component can be quantitatively characterized as a function of the nonlinearity, and excitation parameters, thereby providing an alternate route to inversely measure nonlinear material properties from numerical or experimental data. This is demonstrated numerically by considering granular chains interacting through generic power law relations, and obtaining the imposed power law through the inverse problem set up for the corresponding periodic structure. The other important effect of nonlinearity is the generation of harmonics. The interplay of dispersion and nonlinearity in the generation of harmonics gives rise to secondary wave packets with multiple characteristics. Particularly, one of the nonlinearly generated harmonic features characteristics that conform to the dispersive properties of the corresponding linear periodic structure. This provides an opportunity to engineer wave characteristics through the geometric and topological design of the unit cell, and results in the ability to activate complementary functionalities at low frequencies of excitation - effects seldom observed in linear periodic structures. The ability to design adaptive and tunable phononic switches are demonstrated using a number of periodic structure configurations which cover the entire gamut of nonlinear mechanisms realizable in elastic solids. Finally, an experimental characterization of harmonic generation is presented for the case of a nonlinear lattice by employing 3D Scanning Laser Doppler Vibrometry.