Magnetoelectric random access memory (MERAM) has emerged as a promising new class of non-volatile solid-state memory device. It offers nondestructive reading along with low power consumption during the write operation. A common implementation of MERAM involves use of multiferroic tunneling junctions (MFTJs), which besides offering non-volatility are both electrically and magnetically tunable. Fundamentally, a MFTJ consists of a heterostructure of an ultrathin multiferroic or ferroelectric material as the active tunneling barrier sandwiched between ferromagnetic electrodes. Thereby, the MFTJ exhibits both tunnel electroresistance (TER) and tunnel magnetoresistance (TMR) effects with application of an electric and magnetic field, respectively. In this thesis work, we have developed two-dimensional (2D) thin-film multiferroic heterostructure METJ prototypes consisting of ultrathin ferroelectric BaTiO3 (BTO) layer and a conducting ferromagnetic La0.67Sr0.33MnO3 (LSMO) electrode. The heteroepitaxial films are grown using the pulsed laser deposition (PLD) technique. This oxide heterostructure offers the opportunity to study the nano-scale details of the tunnel electroresistance (TER) effect using scanning probe microscopy techniques. We performed the measurements using the MFP-3D (Asylum Research) scanning probe microscope. The ultrathin BTO films (1.2 - 2.0 nm) grown on LSMO electrodes display both ferro- and piezo-electric properties and exhibit large tunnel resistance effect. We have explored the growth and properties of one-dimensional (1D) heterostructures, referred to as multiferoric nanowire (NW) heterostructures. The ferromagnetic/ferroelectric composite heterostructures are grown as sheath layers using PLD on lattice-matched template NWs, e.g. MgO, that are deposited by chemical vapor deposition utilizing the vapor-liquid-solid (VLS) mechanism. The one-dimensional geometry can substantially overcome the clamping effect of the substrate present in two-dimensional structures because of the reduced volume of the template. This leads to minimum constraint of displacements at the interface and thereby significantly enhances the magnetoelectric (ME) effect. We characterized the nanostructures using scanning electron microscopy (SEM) and transmission electron microscopy (TEM). The results of our studies utilizing multiferroic 2-D thin films and 1-D NW architectures clearly demonstrate the potential of these heterostructures for future device applications, such as in MERAM, data storage, magneto-electric field sensors, etc