1 Introduction

2 Free energy and phase transition  
2.1 Polymer model  
2.2 Some key formulas and notations  
2.3 Quenched free energy  
2.4 Annealed free energy and hierarchy of moments  
2.5 Phase transition  

3 Weak Disorder, Strong Disorder  
3.1 The normalized partition function  
3.2 The self-consistency equation and UI properties in the weak disorder  
3.3 The $\mathbf{L^{2}}$-region  
3.4 Relations between the different critical temperatures  

4 Directional free energy  
4.1 Point-to-point partition function  
4.2 Free energy does not depend on direction  
4.3 Local limit theorem  

5 The replica overlap and localization  
5.1 The compensated Poisson measure and some associated martingales  
5.2 The Doob-Meyer decomposition of ln $Z_{t}$  
5.3 The replica overlap and quenched overlaps    
5.4 Endpoint localization  
5.5 Favorite path and path localization  

6 Formulas for variance and concentration  
6.1 The critical exponents  
6.2 The Clark-Ocone representation  
6.3 The variance formula  

7 Cameron-Martin transform and applications    
7.1 Tilting the polymer  
7.2 Consequences for weak disorder regime  
7.3 Moderate and large deviations at all temperature  

8 Phase diagram in the $(\beta,\nu)$-plane, $d\geq3$  
8.1 Strategy for critical curve estimates  
8.2 Main results  
8.3 Main steps  

9 Complete localization  
9.1 A mean field limit   
9.2 The regime of complete localization  

10 The Intermediate Regime ($d=1$)  
10.1 Introduction  
10.2 Connections between stochastic heat equation(s) and directed polymers  
10.3 Chaos expansions  
10.4 The intermediate regime  
10.5 Convergence in terms of processes of the P2P partition function