Let $M$ be a two dimensional totally real submanifold of class $C^2$ in ${\mathbf{C}}^2$. A continuous map $F:\bar{\Delta }\to {\mathbf{C}}^2$ of the closed unit disk $\bar{\Delta }\subset \mathbf{C}$ into ${\mathbf{C}}^2$ that is holomorphic on the open disk $\Delta$ and maps its boundary $b\Delta$ into $M$ is called an analytic disk with boundary in $M$. Given an initial immersed analytic disk $F^0$ with boundary in $M$, we describe the existence and behavior of analytic disks near $F^0$ with boundaries in small perturbations of $M$ in terms of the homology class of the closed curve $F^0\left(b\Delta \right)$ in $M$. We also prove a regularity theorem...