We present results about large deviations and laws of large numbers for various polymer related quantities. In a completely general setting and strictly positive temperature, we present results about large deviations for directed polymers in random environment. We prove quenched large deviations (and compute the rate functions explicitly) for the exit point of the polymer chain and the polymer chain itself. We also prove existence of the upper tail large deviation rate function for the logarithm of the partition function. In the case where the environment weights have certain log-gamma distributions the computations are tractable and allow us to compute the rate function explicitly. At zero temperature, the polymer model is now called a last passage model. With a particular choice of random weights, the last passage model has an equivalent representation as a particle system called Totally Asymmetric Simple Exclusion Process (TASEP). We prove a hydrodynamic limit for the macroscopic particle density and current for TASEP with spatially inhomogeneous jump rates given by a speed function that may admit discontinuities. The limiting density profiles are described with a variational formula. This formula enables us to compute explicit density profiles even though we have no information about the invariant distributions of the process. In the case of a two-phase flux for which a suitable p.d.e. theory has been developed we also observe that the limit profiles are entropy solutions of the corresponding scalar conservation law with a discontinuous speed function.
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