This survey mainly deals with the tilted Carath{ \'e}odory class by angle $\lambda$ (denoted by $\mathcal{ P}_{ \lambda}$) an element of which maps the unit disc into the tilted right half-plane $\{ w: \Re e^{ i\lambda} w>0\}$. Firstly we will characterize $\mathcal{ P}_{ \lambda}$ from different aspects. In section 3, various estimates of functionals over $\mathcal{ P}_{ \lambda}$ are deduced from the known corresponding estimates of $\mathcal{ P}_{ 0}$ or elementary functional analysis. Finally some subsets of analytic functions related to $\mathcal{ P}_{ \lambda}$ including close-to-convex functions with argument $\lambda$, $\lambda$-spirallike functions, $\lambda$-Robertson functions and analytic functions whose derivative is in $\mathcal{ P}_{ \lambda}$ are also considered as applications
Categories: math.CV