In this work the authors deal with linear second order partial differential operators of the following type $H=partial_{t}-L=partial_{t}-sum_{i,j=1}^{q}a_{ij}(t,x) X_{i}X_{j}-sum_{k=1}^{q}a_{k}(t,x)X_{k}-a_{0}(t,x)$ where $X_{1},X_{2},ldots,X_{q}$ is a system of real Hormander's vector fields in some bounded domain $Omegasubseteqmathbb{R}^{n}$, $A=left{ a_{ij}left( t,xright) right} _{i,j=1}^{q}$ is a real symmetric uniformly positive definite matrix such that $lambda^{-1}vertxivert^{2}leqsum_{i,j=1}^{q}a_{ij}(t,x) xi_{i}xi_{j}leqlambdavertxivert^{2}text{}forallxiinmathbb{R}^{q}, x inOmega,tin(T_{1},T_{2})$ for a suitable constant $lambda>0$ a for some real numbers $T_{1} < T_{2}$. Table of Contents: Introduction. Part I: Operators with constant coefficients: Overview of Part I; Global extension of Hormander's vector fields and geometric properties of the CC-distance; Global extension of the operator $H_{A}$ and existence of a fundamental solution; Uniform Gevray estimates and upper bounds of fundamental solutions for large $dleft(x,yright)$; Fractional integrals and uniform $L^{2}$ bounds of fundamental solutions for large $dleft(x,yright)$; Uniform global upper bounds for fundamental solutions; Uniform lower bounds for fundamental solutions; Uniform upper bounds for the derivatives of the fundamental solutions; Uniform upper bounds on the difference of the fundamental solutions of two operators. Part II: Fundamental solution for operators with Holder continuous coefficients: Assumptions, main results and overview of Part II; Fundamental solution for $H$: the Levi method; The Cauchy problem; Lower bounds for fundamental solutions; Regularity results. Part III: Harnack inequality for operators with Holder continuous coefficients: Overview of Part III; Green function for operators with smooth coefficients on regular domains; Harnack inequality for operators with smooth coefficients; Harnack inequality in the non-smooth case; Epilogue; References. (MEMO/204/961)