## 1 Introduction  
In laser plasma acceleration (LPA) [1], an ultra short laser pulse generates a plasma wave with relativistic phase velocity arising from the ponderomotive force of the pulse. Particles can be accelerated with electric fields up to three orders of magnitude (\(>\)100 GeV/m) [2] larger than conventional radio frequency accelerators, motivating the development of shorter accelerators for high energy physics and light source applications. Along with the high power laser development thanks to chirped pulse amplification [3], significant electron beam acceleration was achieved [4; 5; 6; 7; 8] at the beginning of this century. LPA can provide accelerating gradients from a few hundreds of MeV to several GeV per cm and reasonable beam characteristics (typical current of a few kiloamperes, relative energy spread of the order of 1%, and a normalized emittance of the order of 1\(\pi\) mm mrad) [9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21]. However, all these \"best\" performances are usually not achieved simultaneously and depend on the chosen configuration. Different methods have been proposed to control the plasma wave and electrons injection. The colliding pulse regime [9; 22] high-quality electron bunch with low energy spread can have tuneable energy by changing the laser collision position. Density ramp injection [23] generates well-collimated, short electron bunches with narrow energy spread. Ionization injection [24; 25; 26], with a simple and robust setup and a stable injection mechanism, allows for stable performance over multiple hours of operation, as well as easy tuning of beam parameters through modification of the gas characteristics. LPA, now capable of generating electron beams of several GeV [19; 27], with fs duration [12], excellent emittance at the plasma-vacuum interface (\(\sim\)1 mm mrad), and high peak current (1-10 kA) within the cm scale, drives a large interest for different applications such as future colliders [28], light sources (undulator radiation [29], and free electron laser (FEL) [30; 31; 32; 33; 34; 35]). LPA has to move from the new acceleration concept to prove that the devices can be operated as an accelerator. For such a purpose, the electron beam along a transport line has to be controlled. Specific requirements appear for different applications. For example, the collider application requires being able to stage LPA accelerators, but so far, the energy step is still limited [36; 37] since the transport between the different stages is very challenging. The feasibility of observation of undulator radiation was demonstrated [38; 39; 40; 41] after a first electron beam strong focus, but without evidencing the full properties as currently achieved on the synchrotron radiation light source, which would make it attractive for user applications. For the FEL application, proper beam parameters should be provided in the undulator for the amplification to be possible, requiring a specific phase space manipulation. The application of conventional accelerator techniques for the transport are not straightforward in the LPA case. The characteristics of the beams in longitudinal and transverse phase spaces are very different. In the longitudinal phase space, whereas the conventional accelerators typically operate at ultra-low energy spread, the LPA exhibits ultra-short electron bunches that are likely to be lengthened along the transport. In the transverse phase space, the LPA appears as a diverging (mrad) point source, whereas the conventional accelerator operates with typically 10-100 \(\upmu\)m divergence. Consequently, the chromatic effects associated with the divergence and the energy spread, which are generally negligible on conventional accelerators, appear to play a major role and complicate the transport, where a significant emittance growth can occur [42; 43; 44; 45]. Strategies are proposed to mitigate this, such as an adiabatic matching section [46], tailoring focusing profiles [47] or creating a decreasing plasma gradient after the source [48].