Molecular Computation (MC) is massively parallel computation where data is stored and processed within objects of molecular size. Biomolecular Computation (BMC) is MC using biotechnology techniques, e.g. recombinant DNA operations. In contrast, Quantum Computation (QC) is a type of computation where unitary and measurement operations are executed on linear superpositions of basis states. Both BMC and QC may be executed at the micromolecular scale by a variety of methodologies and technologies. This paper surveys various methods for doing BMC and QC and discusses the considerable theoretical and practical advances in BMC and QC made in the last few years. We compare bounds on key resource such as time, volume (number of molecules times molecular density), energy and error rates achievable, taking particular note of the scalability of these methods with the size of the problem solved. In addition to NP search problems and database search problems, we enumerate a wide variety of further potential practical applications of BMC and QC.We observe that certain problems (e.g., NP search problems), if solved with polynomial time bounds, requires exponentially large volume for BMC, so BMC does not scale well to solve very large NP search problems. However, we describe a number of applications (e.g., search within large data bases and simulation of parallel machines) where the volume grows polynomial.Also, we note that the observation operation of QC, which is a fundamental operation of QC used for obtaining classical output, may potentially suffer from exponentially large volume requirements. Observation operations in quantum physics are generally done by a macroscopic measurement apparatus, and the original formulations of QC implicity assumed that macroscopic measurement apparatus would be used for QC. However, if the measurement apparatus is very small, it will be subject to quantum effects. At this time, no one has demonstrated or proved that the observation operation (for a quantum system with n entangled qubits) can even be approximated within a larger unitary quantum system in volume growing less than exponentially with n. Hence it is unknown whether QC (with the observation operation) scales to even moderate numbers of qubits within small volume. We pose a major open problem in QC: to determine (i.e., provide a formal proof) whether or not observation, in a closed quantum system, can be approximated in small volume: say, growing as a polynomial in the number of qubits.We also discuss techniques for decreasing errors in BMC (e.g., annealing errors) and QC (e.g., decoherence errors), and volume where possible, to insure the scalability of BMC and QC to problems of large input size. In addition, we consider how BMC might be used to assist QC (e.g., to do observation operations for Bulk QC) and also how quantum techniques might be used to assist BMC (e.g., to do exquisite detection of very small quantities of a molecule in solution within a large volume).