This chapter provides an overview of advanced techniques for monitoring the condition of mission-critical railway assets. The safe operation of railways depends on a large number of geographically distributed components, each of which has a low cost when compared to the highly complex arrangements of assets found in other industries, such as rolling mills and chemical plants. Failure of any one of these components usually results in a degradation of service in order to maintain safety, and is thus very costly to modern railway operators, who are required to compensate their customers when delays occur. In this chapter, techniques for industrial condition monitoring are reviewed, highlighting the main approaches and their applicability, advantages, and disadvantages. The chapter first makes some basic definitions of faults, failures, and machine conditions. The analysis of faults through methods such as Fault Tree Analysis and Failure Modes Effects Analysis are examined. The field of fault diagnosis is then reviewed, partitioning into the three main areas: numeric/analytical models, qualitative models, and data/history-based methods. Some of the key approaches within each of these areas will be explained at a high level, compared, and contrasted.