Novel phenomena and functionalities at epitaxial complex oxide heterostructures have been attracting huge scientific attention because of the intriguing fundamental physics as well as potential for technological applications that they embody. Essentially, charge and spin reconstruction at the interface can lead to exotic properties, which are completely different from those inherent to the individual materials, for example, a conductive interface between two insulating materials and interface ferromagnetism in the proximity of an antiferromagnet. The interplay between charge and spin degrees of freedom can be particularly intriguing, leading to a fascinating realm, called multiferroic. In this dissertation, a systematic study is performed on the electronic (charge) and magnetic (spin) interaction/reconstruction across the interface of an all-oxide model heterostructure system consisting of the ferromagnet (FM) La$_{0.7}$Sr$_{0.3}$MnO$_3$ (LSMO) and the multiferroic (ferroelectric and antiferromagnetic) BiFeO$_3$ (BFO). The study demonstrates two pathways of using these exotic interfacial properties to control bulk properties, both the ferroelectricity in BFO and ferromagnetism in LSMO. The journey starts with the growth of high-quality BFO/LSMO heterostructures with unit-cell precision control using reflection high-energy electron diffraction combined with pulsed-laser deposition, providing an important platform for the investigation of electronic and magnetic coupling phenomena across the interface. First, we have observed a novel consequence of the interface electronic interaction due to the so-called ``polar discontinuity'', namely, a built-in electrostatic potential accumulates across the heterointerface, and provides deterministic control of ferroelectric polarization states in thin films. This observation suggests a strong, delocalized effect with important implications for future electronics based on such materials. Secondly, we have revealed a strong magnetic coupling at this interface, manifested in the form of an enhanced coercive field as well as a significant exchange-bias coupling. Based on our x-ray magnetic circular dichroism studies, the origin of the exchange-bias coupling is attributed to a novel ferromagnetic state formed in the antiferromagnetic BFO sublattice at the interface with LSMO. Thirdly, using a field effect geometry, we have proposed a pathway to use an electric field to control the magnetism in LSMO in which the ground state of the interfacial ferromagnetic state is strongly correlated with the ferroelectric polarization. Magnetotransport measurements clearly demonstrate a reversible switch/control between two distinct exchange-bias states by isothermally switching the ferroelectric polarization of BFO. This is an important step towards controlling magnetization with the electric field, which may enable a new class of electrically controllable spintronic devices and provide a new basis for producing electrically controllable spin-polarized currents. Finally, combining experimental results with first-principle and phenomenological model calculations, a microscopic model has been proposed to understand the underlying physics of the magnetoelectric coupling, providing further insights on achieving the electric-field control of magnetism. In summary, our studies on the interfacial electronic and magnetic properties at BFO/LSMO heterointerfaces have revealed a strong interplay between the charge, spin, orbital and lattice degrees of freedom at the interface, which will have important implications for a new pathway to use the interface properties to control bulk functionalities (ferroelectric polarization and ferromagnetic magnetization in this study). Such couplings at the interface may be extended to other oxides and will bring into play remarkable physical concepts to this developing field of complex oxide heterointerfaces.