Biocomputing (also known as molecular or cell computing) is a new interdisciplinary area of science where computer science, chemistry, biology, and physics meet, that promises revolutionary changes in computations in the near future. It aims at developing new devices and computing systems to manipulate information operating at atomic or molecular scale on the basis of biological, chemical, electronic, photonic and/or mechanical principles. In the paper a short overview of biocomputing area and results obtained in the frame of STCU project 4032 “Power and efficiency of natural computing: neural-like P (membrane) systems” (project manager: Prof. Iurie Rogojin; coordinating institution: Institute of Mathematics and Computer Science of the Academy of Sciences of Moldova) are presented. In particular new computational possibilities provided by computing with biomolecules in the following biocomputing models are investigated: Networks of Evolutionary Processors (NEPs) and Hybrid Networks of Evolutionary Processors (HNEPs), Communication P Systems, Transitional P Systems, P Systems with Active Membranes, Polymorphic P Systems, Insertion-Deletion Systems, Insertion-Deletion P Systems with Small Parameters, and Splicing P Systems. Promising results of application of biocomputing methods and results in computer algebra and computer linguistics are presented too.