Physical Unclonable Functions (PUFs) are simple circuits that are ideal for hardware security. Typically, they are used for identifying and authenticating integrated circuits (ICs). In this work, we are interested in a class of delay based PUFs which mainly consist of multiplexers. They are known as multiplexer-based PUFs or MUX PUFs, for short. We are interested in modelling their structure and then, analyzing their performances. Our work can be mainly divided into some key contributions. First, we discuss about the different types of MUX PUFs that we deal with in this work. They are the simple or linear configuration, feed-forward configuration and modified feed-forward configuration. We then, present a typical scheme used for the authentication of these PUFs. However, much of the work concentrates on a modified version of the authentication scheme, where instead of storing a look-up table (LUT) of challenge-response pairs (CRP) in the server, we store a set of delay parameters corresponding to the physical attributes of the MUX PUF. These stored parameters are the delay-differences of the MUX stage and the arbiter delay. We show that MUX PUFs can be modelled using an additive linear delay model. The additive model helps in the computation of an important parameter, known as total delay-difference. Based on the total delay-difference, we can compute two different versions of the output or response: hard-response, which is either a `0' or `1' bit and soft-response, which can take continuous values between 0 and 1. We formulate models for obtaining both these responses. Various metrics used for the evaluation of PUF performance are discussed. The general lab setup used to collect the required PUF data is also discussed. Next, we discuss about the various effects of aging on the performance of MUX PUFs. We extend the linear delay model to include the variations in delay parameters due to aging. The model makes certain assumptions about how noise and aging affect the delay chain (consisting of the multiplexers) and the arbiter. We assume that for a fixed set of conditions, the noise can only cause a constant amount of degradation to the performance of an aging PUF. However, aging which is caused due to undesirable changes like negative bias temperature instability (NBTI), hot carrier injection (HCI) and time dependent dielectric breakdown (TDDB) results in a gradual degradation of performance. That is, the variations due to aging gradually increase with time in contrast to that of noise. In our study, we compare the standalone effects of aging and noise on the PUF. We observe that for the same amount of variation, aging degrades the authentication performance much more than noise. Furthermore, experimental aging data collected from PUFs in our lab suggest that the percent variation in delay parameters can be modelled as a Gaussian distribution. However, there is a small difference in how the percent variations of delay-differences of MUX stages and the arbiter delay are modelled. The former is a zero mean Gaussian, whereas the latter is a positive mean Gaussian with mean and variance both gradually increasing with aging. In addition, the variation in arbiter delay is assumed to be higher than that of delay-differences due to ``asymmetric'' aging in case of arbiter. This happens under unequal aging scenario. Using a Monte-Carlo based simulation for aging, authentication accuracy of the three configurations are studied. We also suggest approaches to improve the authentication accuracy that will increase the lifetime of a PUF. This can be done by either recalibrating the delay parameters or by tuning a threshold based on total delay-difference. Next, we discuss an entropy based approach that can be used to identify whether a MUX is linear or non-linear. The approach is focused on computing the conditional entropy of responses to a set of predefined challenges. The challenge set consists of randomly chosen challenges and their 1-bit neighbors. The entropy is computed across the responses of two 1-bit neighboring challenges. For non-linear MUX PUFs like feed-forward, the method determines the MUX stages which are controlled by internally generated challenge bits as opposed to external challenge bits. This is based on the observation that the conditional entropy for each of these stages is zero. Also, the number of zero conditional entropy values across the MUX stages provide an upper bound on the number of internal arbiters present in the PUF. With the proposed approach, we observe 100% sensitivity and 100% specificity for identifying non-linearity. Furthermore, we show that the proposed approach requires very less number of stable random challenges (about 50) for successfully determining whether a PUF is linear or not for real chips. Our next contribution involves a logistic regression based approach to predict the soft-response for a challenge using the total delay-difference as an input. This approach enables us to determine whether a challenge is stable or not. The approach learns a logistic function based on the total delay-difference which has just 3 parameters. Therefore, this is a simple approach which gives comparable performance against a more complex approach based on artificial neural network (ANN) models. The model demonstrates good sensitivity and precision but poor specificity. Finally, we discuss a bit-flipping algorithm used to convert the unstable challenges to stable challenges. It is based on the idea that a threshold on the total delay-difference can guarantee stability of challenges. The thresholds can be obtained empirically from the probability distributions of the total delay-difference. A straightforward approach is to discard and issue a new random challenge for authentication if the current challenge is unstable. In this paper, we propose a novel bit-flipping based approach in which we claim that by flipping few bits of the original unstable challenge, we can convert it to a stable one with minimal number of bit-flips. By using the algorithm, we are able to transform the most likely unstable challenges to stable ones, typically with 1 bit-flip for linear and modified feed-forward PUFs and 3 bit-flips for the feed-forward PUFs. These bit-flips correspond to the flips in the XOR-ed challenge. We also compare the computation complexities of best, average and worst-case scenarios for the straightforward and proposed approaches. In terms of number of addition operations, the proposed approach has slightly better average-case performance but much better worst-case performance than the straightforward approach.