We experimentally demonstrate a source of nearly pure single photons in arbitrary temporal shapes heralded from a parametric down-conversion (PDC) source at telecom wavelengths. The technology is enabled by the tailored dispersion of in-house fabricated waveguides with shaped pump pulses to directly generate the PDC photons in on-demand temporal shapes. We generate PDC photons in Hermite-Gauss and frequency-binned modes and confirm a minimum purity of 0.81, even for complex temporal shapes.Preparing single photons in pure and controlled spectral-temporal modes is a key requirement for quantum photonic technologies. Diverse applications including quantum-enhanced metrology [1,2], quantum computation [3,4], and quantum encryption [5][6][7] rely on high-contrast interference through stable sources of pure single photons. In addition, widely customisable and precisely controllable temporal-mode shaping is necessary to ensure mode matching between individual sources [8], facilitate coupling between nodes in a quantum network [9], and enable temporal-mode based quantum communication [10] and source mupliplexing [11,12], among other applications. Furthermore, sources with high brightness are essential for scalable performance, and spatially single-mode behaviour is necessary for coupling to optical fibre networks and integrated waveguide devices.Sources based on parametric downconversion (PDC) have granted a simple solution to heralded single-photon generation for decades, but have not yet satisfied all of the above requirements simultaneously. Most PDC sources generate photons with strong spectral correlations which is undesirable for heralded single-photon sources. However, it is possible to minimise the spectral correlation in crystals offering specific dispersion properties along with an adapted pump bandwidth [8,[13][14][15][16][17][18][19][20][21]. This specific dispersion property is linked to the group velocities of the pump and the PDC photons and can be summarised in two categories: matching the group velocity of the pump photon with one of the PDC photons [8,20], or having the group velocity of the pump between the two PDC photons [15,[17][18][19].On the other hand, efficient temporal-mode shaping of the PDC photons is more challenging. Existing methods to create a broadband single photon in an arbitrary temporal mode rely on carving out the desired mode from the original wavepacket as depicted in Fig. 1(a), which can be accurately achieved by temporal or spectral modulation of the photon [22][23][24][25][26]. This method, however, necessarily introduces loss and leads to a reduced rate of pre- * vahid.ansari@uni-paderborn.de With an appropriately designed pump field and group-velocity engineered nonlinear medium, the PDC photons are emitted directly in a desired temporal shape. In both scenarios the purity of heralded single photon rely on the separability of the PDC state in terms of signal and herald fields.pared photons [27] and a low pair-symmetric heralding efficiencies [28]; this poses a practical lim...