1 Introduction
2 Pulsars, Observations, and Timing
2.1 Pulsar properties
2.2 Pulsar observations
2.3 Pulsar timing
$$\begin{array}{c}\phi ={\phi}_{0}+\nu (t{t}_{0})+\frac{1}{2}\dot{\nu}(t{t}_{0}{)}^{2}+...,\end{array}$$  (1) 
$$\begin{array}{c}Bsin\alpha ={\left(\frac{3I\dot{\nu}{c}^{3}}{8{\pi}^{2}{R}^{6}{\nu}^{3}}\right)}^{1/2}\approx 3.2\times {10}^{19}{\left(\frac{\dot{\nu}}{{\nu}^{3}}\right)}^{1/2}G,\end{array}$$  (2) 
$$\begin{array}{c}{\tau}_{c}=\frac{\nu}{2\dot{\nu}}.\end{array}$$  (3) 
2.3.1 Basic transformation
$$\begin{array}{c}t=\tau D/{f}^{2}+{\Delta}_{R\odot}+{\Delta}_{E\odot}{\Delta}_{S\odot}{\Delta}_{R}{\Delta}_{E}{\Delta}_{S}.\end{array}$$  (4) 
2.3.2 Binary pulsars
$$\begin{array}{ccc}uesinu& =& 2\pi \left[\left(\frac{T{T}_{0}}{{P}_{b}}\right)\frac{{\dot{P}}_{b}}{2}{\left(\frac{T{T}_{0}}{{P}_{b}}\right)}^{2}\right],\end{array}$$  (5) 
$$\begin{array}{ccc}{A}_{e}\left(u\right)& =& 2arctan\left[{\left(\frac{1+e}{1e}\right)}^{1/2}tan\frac{u}{2}\right],\end{array}$$  (6) 
$$\begin{array}{ccc}\omega & =& {\omega}_{0}+\left(\frac{{P}_{b}\dot{\omega}}{2\pi}\right){A}_{e}\left(u\right),\end{array}$$  (7) 
$$\begin{array}{ccc}{\Delta}_{R}& =& xsin\omega (cosue(1+{\delta}_{r}\left)\right)+x(1{e}^{2}(1+{\delta}_{\theta}{)}^{2}{)}^{1/2}cos\omega sinu,\end{array}$$  (8) 
$$\begin{array}{ccc}{\Delta}_{E}& =& \gamma sinu,\end{array}$$  (9) 
$$\begin{array}{ccc}{\Delta}_{S}& =& 2rln\left\{1ecosus\left[sin\omega (cosue)+(1{e}^{2}{)}^{1/2}cos\omega sinu\right]\right\}.\end{array}$$  (10) 
3 Tests of GR – Equivalence Principle Violations
$$\begin{array}{c}{c}_{i}=2\frac{\partial ln{m}_{i}}{\partial lnG}\simeq {\left(\frac{2{E}^{grav}}{m{c}^{2}}\right)}_{i},\end{array}$$  (11) 

3.1 Strong Equivalence Principle: Nordtvedt effect
$$\begin{array}{c}\eta =4\beta \gamma 3\frac{10}{3}\xi {\alpha}_{1}+\frac{2}{3}{\alpha}_{2}\frac{2}{3}{\zeta}_{1}\frac{1}{3}{\zeta}_{2}.\end{array}$$  (12) 
$$\begin{array}{ccc}{\left(\frac{{m}_{grav}}{{m}_{inertial}}\right)}_{i}& =& 1+{\Delta}_{i}\end{array}$$ 
$$\begin{array}{ccc}& =& 1+\eta {\left(\frac{{E}^{grav}}{m{c}^{2}}\right)}_{i}+{\eta}^{\prime}{\left(\frac{{E}^{grav}}{m{c}^{2}}\right)}_{i}^{2}+....\end{array}$$  (13) 
$$\begin{array}{c}\mathbf{e}\left(t\right)={\mathbf{e}}_{F}+{\mathbf{e}}_{R}\left(t\right),\end{array}$$  (14) 
$$\begin{array}{c}\left{\mathbf{e}}_{F}\right=\frac{3}{2}\frac{{\Delta}_{net}{\mathbf{g}}_{\perp}}{\dot{\omega}a(2\pi /{P}_{b})},\end{array}$$  (15) 
$$\begin{array}{c}\left{\mathbf{e}}_{F}\right=\frac{1}{2}\frac{{\Delta}_{net}{\mathbf{g}}_{\perp}{c}^{2}}{FGM(2\pi /{P}_{b}{)}^{2}},\end{array}$$  (16) 
$$\begin{array}{c}\left{\mathbf{e}}_{F}\right\le e{\xi}_{1}\left(\theta \right),{\xi}_{1}\left(\theta \right)=\{\begin{array}{cc}1/sin\theta & \text{for}\theta \in [0,\pi /2),\\ 1& \text{for}\theta \in [\pi /2,3\pi /2],\\ 1/sin\theta & \text{for}\theta \in (3\pi /2,2\pi ),\end{array}\end{array}$$  (17) 
$$\begin{array}{c}\left{\mathbf{g}}_{\perp}\right=\mathbf{g}[1(cosicos\lambda +sinisin\lambda sin\Omega {)}^{2}{]}^{1/2},\end{array}$$  (18) 
3.2 Preferredframe effects and nonconservation of momentum
3.2.1 Limits on $\text{}{\hat{\mathit{\alpha}}}_{1}\text{}$
$$\begin{array}{c}\left{\mathbf{e}}_{F}\right=\frac{1}{12}{\hat{\alpha}}_{1}\left\frac{{m}_{1}{m}_{2}}{{m}_{1}+{m}_{2}}\right\frac{\left{w}_{\perp}\right}{{\left[G({m}_{1}+{m}_{2})(2\pi /{P}_{b})\right]}^{1/3}},\end{array}$$  (19) 
3.2.2 Limits on $\text{}{\hat{\mathit{\alpha}}}_{3}\text{}$
$$\begin{array}{c}{\mathbf{a}}_{self}=\frac{1}{3}{\hat{\alpha}}_{3}\frac{{E}^{grav}}{\left(m{c}^{2}\right)}\mathbf{w}\times {\mathbf{\Omega}}_{S}.\end{array}$$  (20) 
$$\begin{array}{c}{\dot{P}}_{{\hat{\alpha}}_{3}}=\frac{P}{c}\hat{\mathbf{n}}\cdot {\mathbf{a}}_{self},\end{array}$$  (21) 
$$\begin{array}{c}{\dot{P}}_{pm}=P{\mu}^{2}\frac{d}{c},\end{array}$$  (22) 
$$\begin{array}{c}{\mathbf{a}}_{self}=\frac{1}{6}{\hat{\alpha}}_{3}{c}_{p}\mathbf{w}\times {\mathbf{\Omega}}_{Sp},\end{array}$$  (23) 
$$\begin{array}{c}\left{\mathbf{e}}_{F}\right={\hat{\alpha}}_{3}\frac{{c}_{p}\mathbf{w}}{24\pi}\frac{{P}_{b}^{2}}{P}\frac{{c}^{2}}{G({m}_{1}+{m}_{2})}sin\beta ,\end{array}$$  (24) 
3.2.3 Limits on $\text{}{\mathit{\zeta}}_{2}\text{}$
$$\begin{array}{c}{\mathbf{a}}_{cm}=({\alpha}_{3}+{\zeta}_{2})\frac{\pi {m}_{1}{m}_{2}({m}_{1}{m}_{2})}{{P}_{b}\left[\right({m}_{1}+{m}_{2}\left)a\right(1{e}^{2}){]}^{3/2}}e{\mathbf{n}}_{p},\end{array}$$  (25) 
$$\begin{array}{c}\ddot{P}=\frac{P}{2}({\alpha}_{3}+{\zeta}_{2}){m}_{2}sini{\left(\frac{2\pi}{{P}_{b}}\right)}^{2}\frac{X(1X)}{(1+X{)}^{2}}\frac{e\dot{\omega}cos\omega}{(1{e}^{2}{)}^{3/2}},\end{array}$$  (26) 
3.3 Strong Equivalence Principle: Dipolar gravitational radiation
$$\begin{array}{c}{\dot{P}}_{bdipole}=\frac{4{\pi}^{2}{G}_{*}}{{c}^{3}{P}_{b}}\frac{{m}_{1}{m}_{2}}{{m}_{1}+{m}_{2}}({\alpha}_{{c}_{1}}{\alpha}_{{c}_{2}}{)}^{2},\end{array}$$  (27) 
3.4 Preferredlocation effects: Variation of Newton's constant
3.4.1 Spin tests
$$\begin{array}{c}{\left(\frac{\dot{P}}{P}\right)}_{\dot{G}}={\left(\frac{\partial lnI}{\partial lnG}\right)}_{N}\frac{\dot{G}}{G},\end{array}$$  (28) 
3.4.2 Orbital decay tests
$$\begin{array}{c}{\left(\frac{{\dot{P}}_{b}}{{P}_{b}}\right)}_{\dot{G}}=2\frac{\dot{G}}{G}.\end{array}$$  (29) 
$$\begin{array}{c}{\left(\frac{{\dot{P}}_{b}}{{P}_{b}}\right)}_{\dot{G}}=\left[2\left(\frac{{m}_{1}{c}_{1}+{m}_{2}{c}_{2}}{{m}_{1}+{m}_{2}}\right)\frac{3}{2}\left(\frac{{m}_{1}{c}_{2}+{m}_{2}{c}_{1}}{{m}_{1}+{m}_{2}}\right)\right]\frac{\dot{G}}{G}.\end{array}$$  (30) 
3.4.3 Changes in the Chandrasekhar mass
4 Tests of GR – StrongField Gravity
4.1 PostKeplerian timing parameters
$$\begin{array}{ccc}\dot{\omega}& =& 3{\left(\frac{{P}_{b}}{2\pi}\right)}^{5/3}\left({T}_{\odot}M{)}^{2/3}\right(1{e}^{2}{)}^{1},\end{array}$$  (31) 
$$\begin{array}{ccc}\gamma & =& e{\left(\frac{{P}_{b}}{2\pi}\right)}^{1/3}{T}_{\odot}^{2/3}{M}^{4/3}{m}_{2}({m}_{1}+2{m}_{2}),\end{array}$$  (32) 
$$\begin{array}{ccc}{\dot{P}}_{b}& =& \frac{192\pi}{5}{\left(\frac{{P}_{b}}{2\pi}\right)}^{5/3}\left(1+\frac{73}{24}{e}^{2}+\frac{37}{96}{e}^{4}\right)(1{e}^{2}{)}^{7/2}{T}_{\odot}^{5/3}{m}_{1}{m}_{2}{M}^{1/3},\end{array}$$  (33) 
$$\begin{array}{ccc}r& =& {T}_{\odot}{m}_{2},\end{array}$$  (34) 
$$\begin{array}{ccc}s& =& x{\left(\frac{{P}_{b}}{2\pi}\right)}^{2/3}{T}_{\odot}^{1/3}{M}^{2/3}{m}_{2}^{1}.\end{array}$$  (35) 
4.2 The original system: PSR B1913+16

4.3 PSR B1534+12 and other binary pulsars

4.4 Combined binarypulsar tests
4.5 Independent geometrical information: PSR J04374715
$$\begin{array}{c}x\left(t\right)={x}_{0}\left[1+\frac{coti}{d}{\mathbf{r}}_{\oplus}\left(t\right)\cdot {\mathbf{\Omega}}^{\prime}\right],\end{array}$$  (36) 
$$\begin{array}{c}{\dot{x}}_{pm}=xcoti\mu \cdot {\mathbf{\Omega}}^{\prime},\end{array}$$  (37) 
4.6 Spinorbit coupling and geodetic precession
$$\begin{array}{c}\frac{d{\mathbf{S}}_{1}}{dt}={\mathbf{\Omega}}_{1}^{spin}\times {\mathbf{S}}_{1},\end{array}$$  (38) 
$$\begin{array}{c}{\Omega}_{1}^{spin}=\frac{1}{2}{\left(\frac{{P}_{b}}{2\pi}\right)}^{5/3}\frac{{m}_{2}(4{m}_{1}+3{m}_{2})}{(1{e}^{2})({m}_{1}+{m}_{2}{)}^{4/3}}{T}_{\odot}^{2/3},\end{array}$$  (39) 
5 Conclusions and Future Prospects
6 Acknowledgements
Note: The reference version of this article is published by Living Reviews in Relativity