Applying Fourier-transform infrared (FTIR) spectroscopy (or related technologies such as Raman spectroscopy) to biological questions (defined as biospectroscopy) is relatively novel. Potential fields of application include cytological, histological and microbial studies. This potentially provides a rapid and non-destructive approach to clinical diagnosis. Its increase in application is primarily a consequence of developing instrumentation along with computational techniques. In the coming decades, biospectroscopy is likely to become a common tool in the screening or diagnostic laboratory, or even in the general practitioner's clinic. Despite many advances in the biological application of FTIR spectroscopy, there remain challenges in sample preparation, instrumentation and data handling. We focus on the latter, where we identify in the reviewed literature, the existence of four main study goals: Pattern Finding; Biomarker Identification; Imaging; and, Diagnosis. These can be grouped into two frameworks: Exploratory; and, Diagnostic. Existing techniques in Quality Control, Pre-processing, Feature Extraction, Clustering, and Classification are critically reviewed. An aspect that is often visited is that of method choice. Based on the state-of-art, we claim that in the near future research should be focused on the challenges of dataset standardization; building information systems; development and validation of data analysis tools; and, technology transfer. A diagnostic case study using a real-world dataset is presented as an illustration. Many of the methods presented in this review are Machine Learning and Statistical techniques that are extendable to other forms of computer-based biomedical analysis, including mass spectrometry and magnetic resonance.